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Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\mathbb{Z}_{n}$ -graph.
Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.
Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.
Corollary 3.5. For $n\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.
Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected:
By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible.
Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds).
Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\in I m(A)\cap K e r(A-\lambda I)$ , then $B(B x)=\lambda(B x)$ and $A B x=$
$-(1+\lambda)x$ .
Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
|
<html><body>
<p data-bbox="125 110 487 209">Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\mathbb{Z}_{n}$ -graph. </p>
<p data-bbox="124 232 486 261">Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. </p>
<p data-bbox="136 270 417 299">Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. </p>
<p data-bbox="125 321 486 351">Corollary 3.5. For $n\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. </p>
<p data-bbox="124 369 486 398">Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected: </p>
<div class="image" data-bbox="126 424 209 477"><img data-bbox="126 424 209 477"/></div>
<p data-bbox="124 515 487 545">By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. </p>
<p data-bbox="124 557 486 587">Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds). </p>
<p data-bbox="124 595 486 653">Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\in I m(A)\cap K e r(A-\lambda I)$ , then $B(B x)=\lambda(B x)$ and $A B x=$
$-(1+\lambda)x$ . </p>
<p data-bbox="135 660 450 676">Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so </p>
<div class="equation" data-bbox="205 686 405 700">$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$</div>
</body></html>
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0003047v1
| 6 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\\mathbb{Z}_{n}$ -graph. ", "page_idx": 6}, {"type": "text", "text": "Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. ", "page_idx": 6}, {"type": "text", "text": "Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. ", "page_idx": 6}, {"type": "text", "text": "Corollary 3.5. For $n\\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. ", "page_idx": 6}, {"type": "text", "text": "Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected: ", "page_idx": 6}, {"type": "image", "img_path": "images/b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. ", "page_idx": 6}, {"type": "text", "text": "Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds). ", "page_idx": 6}, {"type": "text", "text": "Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\\in I m(A)\\cap K e r(A-\\lambda I)$ , then $B(B x)=\\lambda(B x)$ and $A B x=$ \n$-(1+\\lambda)x$ . ", "page_idx": 6}, {"type": "text", "text": "Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so ", "page_idx": 6}, {"type": "equation", "text": "$$\nA+A^{2}+A B A=B+B^{2}+B A B=0.\n$$", "text_format": "latex", "page_idx": 6}]
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It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [222, 90, 389, 101], "lines": [{"bbox": [223, 92, 388, 102], "spans": [{"bbox": [223, 92, 388, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 100], "lines": [{"bbox": [480, 93, 486, 102], "spans": [{"bbox": [480, 93, 486, 102], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 110, 487, 209], "lines": [{"bbox": [137, 112, 486, 127], "spans": [{"bbox": [137, 112, 486, 127], "score": 1.0, "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 140], "spans": [{"bbox": [126, 127, 435, 140], "score": 1.0, "content": "totally disconnected, there is an edge joining some vertices ", "type": "text"}, {"bbox": [435, 128, 445, 137], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 127, 473, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [473, 128, 482, 137], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [482, 127, 486, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [125, 140, 487, 155], "spans": [{"bbox": [125, 140, 487, 155], "score": 1.0, "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "type": "text"}], "index": 2}, {"bbox": [126, 155, 484, 168], "spans": [{"bbox": [126, 155, 349, 168], "score": 1.0, "content": "3.3 implies that there is an edge between ", "type": "text"}, {"bbox": [349, 156, 359, 165], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 155, 474, 168], "score": 1.0, "content": " and any neighbor of ", "type": "text"}, {"bbox": [475, 156, 484, 165], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 9}], "index": 3}, {"bbox": [126, 169, 486, 182], "spans": [{"bbox": [126, 169, 270, 182], "score": 1.0, "content": "which is not a neighbor of ", "type": "text"}, {"bbox": [270, 171, 279, 180], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 169, 486, 182], "score": 1.0, "content": ". It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3, "bbox_fs": [124, 112, 487, 212]}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [124, 234, 487, 262]}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [138, 272, 415, 299]}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5, "bbox_fs": [126, 324, 487, 352]}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [125, 372, 485, 399]}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [126, 518, 486, 546]}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 559, 485, 588]}, {"type": "list", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22, "is_list_end_line": true}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24, "is_list_start_line": true, "is_list_end_line": true}], "index": 22.5, "bbox_fs": [125, 596, 487, 654]}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [138, 664, 448, 678]}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}]}
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[{"type": "text", "bbox": [125, 110, 487, 209], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices and . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between and any neighbor of which is not a neighbor of . It follows inductively that there is an edge joining to every vertex which is not a neighbor of . Then (c) holds, because the full friendship graph is a -graph.", "index": 0}, {"type": "text", "bbox": [124, 232, 486, 261], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.", "index": 1}, {"type": "text", "bbox": [136, 270, 417, 299], "content": "Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.", "index": 2}, {"type": "text", "bbox": [125, 321, 486, 351], "content": "Corollary 3.5. For , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.", "index": 3}, {"type": "text", "bbox": [124, 369, 486, 398], "content": "Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:", "index": 4}, {"type": "image", "bbox": [126, 424, 209, 477], "content": "", "index": 5}, {"type": "text", "bbox": [124, 515, 487, 545], "content": "By [5], Lemmas 6.2 and 6.3, every representation of of corank 2 and dimension at least 4, which has this friendship graph, is reducible.", "index": 6}, {"type": "text", "bbox": [124, 557, 486, 587], "content": "Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).", "index": 7}, {"type": "list", "bbox": [124, 595, 486, 653], "content": "", "index": 8}, {"type": "text", "bbox": [135, 660, 450, 676], "content": "Proof. (a). By lemma 3.1, and are not true friends, so", "index": 9}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "content": "", "index": 10}]
|
[{"bbox": [137, 112, 486, 127], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 140], "content": "totally disconnected, there is an edge joining some vertices and .", "parent_index": 0, "line_index": 1}, {"bbox": [125, 140, 487, 155], "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "parent_index": 0, "line_index": 2}, {"bbox": [126, 155, 484, 168], "content": "3.3 implies that there is an edge between and any neighbor of", "parent_index": 0, "line_index": 3}, {"bbox": [126, 169, 486, 182], "content": "which is not a neighbor of . It follows inductively that there is an", "parent_index": 0, "line_index": 4}, {"bbox": [126, 183, 484, 196], "content": "edge joining to every vertex which is not a neighbor of . Then (c)", "parent_index": 0, "line_index": 5}, {"bbox": [124, 195, 404, 212], "content": "holds, because the full friendship graph is a -graph.", "parent_index": 0, "line_index": 6}, {"bbox": [124, 234, 487, 250], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "parent_index": 1, "line_index": 0}, {"bbox": [126, 250, 365, 262], "content": "chain, if the only edges are between neighbors.", "parent_index": 1, "line_index": 1}, {"bbox": [138, 272, 391, 286], "content": "Case (b) of the above theorem can be restated as", "parent_index": 2, "line_index": 0}, {"bbox": [140, 288, 415, 299], "content": "(b) The full friendship graph contains the chain graph.", "parent_index": 2, "line_index": 1}, {"bbox": [126, 324, 487, 339], "content": "Corollary 3.5. For , the friendship graph and the full friendship", "parent_index": 3, "line_index": 0}, {"bbox": [126, 339, 439, 352], "content": "graph are either totally disconnected (no edges) or connected.", "parent_index": 3, "line_index": 1}, {"bbox": [125, 372, 485, 386], "content": "Remark 3.6. For there is a friendship graph which is neither", "parent_index": 4, "line_index": 0}, {"bbox": [126, 386, 308, 399], "content": "totally disconnected nor connected:", "parent_index": 4, "line_index": 1}, {"bbox": [137, 518, 486, 532], "content": "By [5], Lemmas 6.2 and 6.3, every representation of of corank 2", "parent_index": 6, "line_index": 0}, {"bbox": [126, 532, 485, 546], "content": "and dimension at least 4, which has this friendship graph, is reducible.", "parent_index": 6, "line_index": 1}, {"bbox": [136, 559, 485, 575], "content": "Now consider the case when the friendship graph is totally discon-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 574, 394, 588], "content": "nected (that is, statement of theorem 3.4 holds).", "parent_index": 7, "line_index": 1}, {"bbox": [125, 596, 443, 613], "content": "Lemma 3.7. If and are neighbors and not friends then:", "parent_index": 8, "line_index": 0}, {"bbox": [140, 612, 292, 624], "content": "(a) ; .", "parent_index": 8, "line_index": 1}, {"bbox": [138, 626, 487, 640], "content": "If , then and", "parent_index": 8, "line_index": 2}, {"bbox": [126, 639, 182, 654], "content": ".", "parent_index": 8, "line_index": 3}, {"bbox": [138, 664, 448, 678], "content": "Proof. (a). By lemma 3.1, and are not true friends, so", "parent_index": 9, "line_index": 0}]
|
[{"bbox": [126, 424, 209, 477], "content": "", "parent_index": 5, "subtype": "body"}]
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[{"bbox": [435, 128, 445, 137], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [473, 128, 482, 137], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [349, 156, 359, 165], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [475, 156, 484, 165], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [270, 171, 279, 180], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [191, 184, 201, 193], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [424, 184, 433, 193], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [353, 198, 367, 209], "content": "\\mathbb{Z}_{n}", "parent_index": 0, "subtype": "inline"}, {"bbox": [234, 326, 263, 337], "content": "n\\neq4", "parent_index": 3, "subtype": "inline"}, {"bbox": [225, 374, 256, 383], "content": "n=4", "parent_index": 4, "subtype": "inline"}, {"bbox": [409, 520, 423, 530], "content": "B_{4}", "parent_index": 6, "subtype": "inline"}, {"bbox": [262, 575, 278, 588], "content": "(a)", "parent_index": 7, "subtype": "inline"}, {"bbox": [213, 600, 222, 608], "content": "A", "parent_index": 8, "subtype": "inline"}, {"bbox": [248, 600, 258, 608], "content": "B", "parent_index": 8, "subtype": "inline"}, {"bbox": [158, 613, 219, 623], "content": "A^{2}B=A B^{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [227, 613, 289, 622], "content": "B A^{2}=B^{2}A", "parent_index": 8, "subtype": "inline"}, {"bbox": [138, 627, 154, 639], "content": "(b)", "parent_index": 8, "subtype": "inline"}, {"bbox": [169, 627, 305, 639], "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "parent_index": 8, "subtype": "inline"}, {"bbox": [338, 627, 421, 640], "content": "B(B x)=\\lambda(B x)", "parent_index": 8, "subtype": "inline"}, {"bbox": [447, 627, 487, 638], "content": "A B x=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 640, 178, 653], "content": "-(1+\\lambda)x", "parent_index": 8, "subtype": "inline"}, {"bbox": [282, 665, 291, 675], "content": "A", "parent_index": 9, "subtype": "inline"}, {"bbox": [317, 665, 327, 675], "content": "B", "parent_index": 9, "subtype": "inline"}, {"bbox": [205, 686, 405, 700], "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "parent_index": 10, "subtype": "interline"}]
|
[]
|
|
Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives
$$
A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.
$$
Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ .
$$
B(B x)=B^{2}A y=B A^{2}y=B A x=\lambda B x,
$$
and
$$
0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\lambda)x+A B x.
$$
Thus, $A B x=-(1+\lambda)x$ .
Theorem 3.8. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ , $(n\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\leq n-1$ .
Proof. If $A_{i}=0$ , $\rho$ is a trivial representation and $r=1$
If $A_{i}\neq0$ , choose an eigenvalue $\lambda$ for $A_{1}$ and a non-zero vector
$$
x_{1}\in I m(A_{1})\cap K e r(A_{1}-\lambda I).
$$
Set $x_{2}=A_{2}x_{1}$ $\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$ By induction and lemma 3.7 (b) $x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$ .
Let $x_{i}\,=\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\;=\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors,
$$
A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\lambda)x_{i-1},\ \ i=2,\ldots,n-1,
$$
$$
A_{i}x_{i}=\lambda x_{i},\;\;\;i=1,\ldots,n-1,
$$
$$
A_{i+1}x_{i}=x_{i+1},\;\;\;i=1,\ldots,n-2,
$$
and
$$
A_{j}x_{i}=A_{j}A_{i}y_{i}=0\;\;j\neq i-1,i,i+1.
$$
Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\le n-1$ , since $\rho$ is irreducible.
Corollary 3.9. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be irreducible, where $r=d i m V\ge$ $n$ , $n\neq4$ .
Then the associated friendship graph is connected.
Proof. By corollary 3.5 the friendship graph of $\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected.
Corollary 3.10. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ .
Then $r=d i m V\leq(n-1)(k-1)+1$ .
In particular, for $k=2$ , $r=d i m V=n$ , where $V=\mathbb{C}^{n}$ .
|
<html><body>
<p data-bbox="123 110 487 138">Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives </p>
<div class="equation" data-bbox="177 144 432 158">$$
A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.
$$</div>
<p data-bbox="125 161 430 176">Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ . </p>
<div class="equation" data-bbox="200 196 408 210">$$
B(B x)=B^{2}A y=B A^{2}y=B A x=\lambda B x,
$$</div>
<p data-bbox="125 215 147 227">and </p>
<div class="equation" data-bbox="149 233 461 249">$$
0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\lambda)x+A B x.
$$</div>
<p data-bbox="124 251 255 266">Thus, $A B x=-(1+\lambda)x$ . </p>
<p data-bbox="125 285 486 327">Theorem 3.8. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ , $(n\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\leq n-1$ . </p>
<p data-bbox="138 334 425 347">Proof. If $A_{i}=0$ , $\rho$ is a trivial representation and $r=1$ </p>
<p data-bbox="135 349 460 362">If $A_{i}\neq0$ , choose an eigenvalue $\lambda$ for $A_{1}$ and a non-zero vector </p>
<div class="equation" data-bbox="228 369 381 382">$$
x_{1}\in I m(A_{1})\cap K e r(A_{1}-\lambda I).
$$</div>
<p data-bbox="124 385 514 413">Set $x_{2}=A_{2}x_{1}$ $\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$ By induction and lemma 3.7 (b) $x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$ . </p>
<p data-bbox="125 414 487 441">Let $x_{i}\,=\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\;=\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors, </p>
<div class="equation" data-bbox="163 448 446 462">$$
A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\lambda)x_{i-1},\ \ i=2,\ldots,n-1,
$$</div>
<div class="equation" data-bbox="229 468 379 480">$$
A_{i}x_{i}=\lambda x_{i},\;\;\;i=1,\ldots,n-1,
$$</div>
<div class="equation" data-bbox="222 484 388 497">$$
A_{i+1}x_{i}=x_{i+1},\;\;\;i=1,\ldots,n-2,
$$</div>
<p data-bbox="125 498 147 511">and </p>
<div class="equation" data-bbox="209 513 399 528">$$
A_{j}x_{i}=A_{j}A_{i}y_{i}=0\;\;j\neq i-1,i,i+1.
$$</div>
<p data-bbox="124 528 488 556">Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\le n-1$ , since $\rho$ is irreducible. </p>
<p data-bbox="124 561 493 590">Corollary 3.9. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be irreducible, where $r=d i m V\ge$ $n$ , $n\neq4$ . </p>
<p data-bbox="140 590 395 604">Then the associated friendship graph is connected. </p>
<p data-bbox="124 609 486 638">Proof. By corollary 3.5 the friendship graph of $\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. </p>
<p data-bbox="124 643 486 672">Corollary 3.10. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ . </p>
<p data-bbox="136 672 422 701">Then $r=d i m V\leq(n-1)(k-1)+1$ .
In particular, for $k=2$ , $r=d i m V=n$ , where $V=\mathbb{C}^{n}$ . </p>
</body></html>
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0003047v1
| 7 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives ", "page_idx": 7}, {"type": "equation", "text": "$$\nA B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ . ", "page_idx": 7}, {"type": "equation", "text": "$$\nB(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\n0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A B x=-(1+\\lambda)x$ . ", "page_idx": 7}, {"type": "text", "text": "Theorem 3.8. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ , $(n\\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\\leq n-1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. If $A_{i}=0$ , $\\rho$ is a trivial representation and $r=1$ ", "page_idx": 7}, {"type": "text", "text": "If $A_{i}\\neq0$ , choose an eigenvalue $\\lambda$ for $A_{1}$ and a non-zero vector ", "page_idx": 7}, {"type": "equation", "text": "$$\nx_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Set $x_{2}=A_{2}x_{1}$ $\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}$ By induction and lemma 3.7 (b) $x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)$ . ", "page_idx": 7}, {"type": "text", "text": "Let $x_{i}\\,=\\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\\;=\\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors, ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\\le n-1$ , since $\\rho$ is irreducible. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.9. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be irreducible, where $r=d i m V\\ge$ $n$ , $n\\neq4$ . ", "page_idx": 7}, {"type": "text", "text": "Then the associated friendship graph is connected. ", "page_idx": 7}, {"type": "text", "text": "Proof. By corollary 3.5 the friendship graph of $\\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.10. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ . ", "page_idx": 7}, {"type": "text", "text": "Then $r=d i m V\\leq(n-1)(k-1)+1$ . \nIn particular, for $k=2$ , $r=d i m V=n$ , where $V=\\mathbb{C}^{n}$ . ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [125, 93, 132, 103], "spans": [{"bbox": [125, 93, 132, 103], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [125, 113, 486, 141]}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3, "bbox_fs": [126, 163, 428, 178]}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [125, 216, 147, 229]}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [126, 253, 254, 268]}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [124, 287, 486, 329]}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11, "bbox_fs": [137, 336, 426, 349]}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [137, 349, 458, 363]}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [125, 386, 513, 415]}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [126, 413, 486, 444]}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [125, 499, 147, 513]}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 529, 486, 557]}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [126, 563, 493, 592]}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [139, 592, 393, 605]}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [126, 612, 484, 639]}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 645, 486, 674]}, {"type": "list", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32, "is_list_end_line": true}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33, "is_list_start_line": true, "is_list_end_line": true}], "index": 32.5, "bbox_fs": [137, 673, 420, 700]}]}
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[{"type": "text", "bbox": [123, 110, 487, 138], "content": "Multiplying the left hand side on the right by and the right hand side on the left by gives", "index": 0}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "content": "", "index": 1}, {"type": "text", "bbox": [125, 161, 430, 176], "content": "Thus, ; by a symmetric argument .", "index": 2}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "content": "", "index": 3}, {"type": "text", "bbox": [125, 215, 147, 227], "content": "and", "index": 4}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "content": "", "index": 5}, {"type": "text", "bbox": [124, 251, 255, 266], "content": "Thus, .", "index": 6}, {"type": "text", "bbox": [125, 285, 486, 327], "content": "Theorem 3.8. Let , ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then .", "index": 7}, {"type": "text", "bbox": [138, 334, 425, 347], "content": "Proof. If , is a trivial representation and", "index": 8}, {"type": "text", "bbox": [135, 349, 460, 362], "content": "If , choose an eigenvalue for and a non-zero vector", "index": 9}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "content": "", "index": 10}, {"type": "text", "bbox": [124, 385, 514, 413], "content": "Set By induction and lemma 3.7 (b) .", "index": 11}, {"type": "text", "bbox": [125, 414, 487, 441], "content": "Let . Then by lemma 3.7 (b) and the fact that , if and are not neighbors,", "index": 12}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "content": "", "index": 13}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "content": "", "index": 14}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "content": "", "index": 15}, {"type": "text", "bbox": [125, 498, 147, 511], "content": "and", "index": 16}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "content": "", "index": 17}, {"type": "text", "bbox": [124, 528, 488, 556], "content": "Thus is invariant under . Hence , since is irreducible.", "index": 18}, {"type": "text", "bbox": [124, 561, 493, 590], "content": "Corollary 3.9. Let be irreducible, where , .", "index": 19}, {"type": "text", "bbox": [140, 590, 395, 604], "content": "Then the associated friendship graph is connected.", "index": 20}, {"type": "text", "bbox": [124, 609, 486, 638], "content": "Proof. By corollary 3.5 the friendship graph of is either totally disconnected or connected. By theorem 3.8 it is not disconnected.", "index": 21}, {"type": "text", "bbox": [124, 643, 486, 672], "content": "Corollary 3.10. Let be irreducible, where , . Suppose , where rank .", "index": 22}, {"type": "list", "bbox": [136, 672, 422, 701], "content": "", "index": 23}]
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[{"bbox": [126, 113, 486, 127], "content": "Multiplying the left hand side on the right by and the right hand", "parent_index": 0, "line_index": 0}, {"bbox": [125, 126, 262, 141], "content": "side on the left by gives", "parent_index": 0, "line_index": 1}, {"bbox": [126, 163, 428, 178], "content": "Thus, ; by a symmetric argument .", "parent_index": 2, "line_index": 0}, {"bbox": [125, 216, 147, 229], "content": "and", "parent_index": 4, "line_index": 0}, {"bbox": [126, 253, 254, 268], "content": "Thus, .", "parent_index": 6, "line_index": 0}, {"bbox": [124, 287, 486, 303], "content": "Theorem 3.8. Let , ) be an irreducible rep-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 303, 485, 316], "content": "resentation, whose associated friendship graph is totally disconnected.", "parent_index": 7, "line_index": 1}, {"bbox": [127, 316, 255, 329], "content": "Then .", "parent_index": 7, "line_index": 2}, {"bbox": [137, 336, 426, 349], "content": "Proof. If , is a trivial representation and", "parent_index": 8, "line_index": 0}, {"bbox": [137, 349, 458, 363], "content": "If , choose an eigenvalue for and a non-zero vector", "parent_index": 9, "line_index": 0}, {"bbox": [125, 386, 513, 403], "content": "Set", "parent_index": 11, "line_index": 0}, {"bbox": [126, 401, 445, 415], "content": "By induction and lemma 3.7 (b) .", "parent_index": 11, "line_index": 1}, {"bbox": [136, 413, 486, 430], "content": "Let . Then by lemma 3.7 (b) and the fact that", "parent_index": 12, "line_index": 0}, {"bbox": [126, 429, 324, 444], "content": ", if and are not neighbors,", "parent_index": 12, "line_index": 1}, {"bbox": [125, 499, 147, 513], "content": "and", "parent_index": 16, "line_index": 0}, {"bbox": [137, 529, 486, 544], "content": "Thus is invariant under . Hence , since is", "parent_index": 18, "line_index": 0}, {"bbox": [125, 544, 184, 557], "content": "irreducible.", "parent_index": 18, "line_index": 1}, {"bbox": [126, 563, 493, 579], "content": "Corollary 3.9. Let be irreducible, where", "parent_index": 19, "line_index": 0}, {"bbox": [126, 578, 173, 592], "content": ", .", "parent_index": 19, "line_index": 1}, {"bbox": [139, 592, 393, 605], "content": "Then the associated friendship graph is connected.", "parent_index": 20, "line_index": 0}, {"bbox": [137, 612, 484, 627], "content": "Proof. By corollary 3.5 the friendship graph of is either totally", "parent_index": 21, "line_index": 0}, {"bbox": [126, 626, 464, 639], "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "parent_index": 21, "line_index": 1}, {"bbox": [126, 645, 486, 660], "content": "Corollary 3.10. Let be irreducible, where", "parent_index": 22, "line_index": 0}, {"bbox": [126, 660, 453, 674], "content": ", . Suppose , where rank .", "parent_index": 22, "line_index": 1}, {"bbox": [138, 673, 333, 689], "content": "Then .", "parent_index": 23, "line_index": 0}, {"bbox": [137, 688, 420, 700], "content": "In particular, for , , where .", "parent_index": 23, "line_index": 1}]
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[]
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[{"bbox": [371, 115, 381, 124], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [223, 128, 232, 137], "content": "A", "parent_index": 0, "subtype": "inline"}, {"bbox": [177, 144, 432, 158], "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "parent_index": 1, "subtype": "interline"}, {"bbox": [159, 165, 221, 175], "content": "A^{2}B=A B^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [363, 165, 425, 175], "content": "B A^{2}=B^{2}A", "parent_index": 2, "subtype": "inline"}, {"bbox": [200, 196, 408, 210], "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "parent_index": 3, "subtype": "interline"}, {"bbox": [149, 233, 461, 249], "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "parent_index": 5, "subtype": "interline"}, {"bbox": [159, 255, 252, 267], "content": "A B x=-(1+\\lambda)x", "parent_index": 6, "subtype": "inline"}, {"bbox": [230, 288, 324, 301], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 7, "subtype": "inline"}, {"bbox": [333, 288, 368, 301], "content": "(n\\geq2,", "parent_index": 7, "subtype": "inline"}, {"bbox": [156, 318, 251, 329], "content": "r=d i m V\\leq n-1", "parent_index": 7, "subtype": "inline"}, {"bbox": [192, 338, 226, 348], "content": "A_{i}=0", "parent_index": 8, "subtype": "inline"}, {"bbox": [232, 341, 239, 349], "content": "\\rho", "parent_index": 8, "subtype": "inline"}, {"bbox": [398, 338, 426, 347], "content": "r=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [149, 352, 183, 363], "content": "A_{i}\\neq0", "parent_index": 9, "subtype": "inline"}, {"bbox": [300, 352, 307, 360], "content": "\\lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [328, 352, 342, 362], "content": "A_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [228, 369, 381, 382], "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "parent_index": 10, "subtype": "interline"}, {"bbox": [145, 389, 197, 400], "content": "x_{2}=A_{2}x_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [193, 389, 513, 401], "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "parent_index": 11, "subtype": "inline"}, {"bbox": [295, 402, 442, 415], "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "parent_index": 11, "subtype": "inline"}, {"bbox": [160, 417, 212, 428], "content": "x_{i}\\,=\\,A_{i}y_{i}", "parent_index": 12, "subtype": "inline"}, {"bbox": [444, 417, 486, 429], "content": "A_{i}A_{j}\\;=\\;", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 431, 173, 443], "content": "A_{j}A_{i}=0", "parent_index": 12, "subtype": "inline"}, {"bbox": [191, 432, 195, 440], "content": "i", "parent_index": 12, "subtype": "inline"}, {"bbox": [221, 432, 227, 442], "content": "j", "parent_index": 12, "subtype": "inline"}, {"bbox": [163, 448, 446, 462], "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "parent_index": 13, "subtype": "interline"}, {"bbox": [229, 468, 379, 480], "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "parent_index": 14, "subtype": "interline"}, {"bbox": [222, 484, 388, 497], "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "parent_index": 15, "subtype": "interline"}, {"bbox": [209, 513, 399, 528], "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "parent_index": 17, "subtype": "interline"}, {"bbox": [168, 532, 177, 541], "content": "U", "parent_index": 18, "subtype": "inline"}, {"bbox": [276, 531, 291, 542], "content": "B_{n}", "parent_index": 18, "subtype": "inline"}, {"bbox": [333, 530, 430, 542], "content": "r=d i m U\\le n-1", "parent_index": 18, "subtype": "inline"}, {"bbox": [466, 533, 473, 543], "content": "\\rho", "parent_index": 18, "subtype": "inline"}, {"bbox": [231, 565, 320, 577], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 19, "subtype": "inline"}, {"bbox": [427, 565, 493, 577], "content": "r=d i m V\\ge", "parent_index": 19, "subtype": "inline"}, {"bbox": [126, 583, 133, 588], "content": "n", "parent_index": 19, "subtype": "inline"}, {"bbox": [140, 580, 169, 591], "content": "n\\neq4", "parent_index": 19, "subtype": "inline"}, {"bbox": [394, 614, 401, 625], "content": "\\rho", "parent_index": 21, "subtype": "inline"}, {"bbox": [241, 646, 344, 659], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "parent_index": 22, "subtype": "inline"}, {"bbox": [463, 647, 486, 658], "content": "r=", "parent_index": 22, "subtype": "inline"}, {"bbox": [126, 661, 178, 672], "content": "d i m V\\geq n", "parent_index": 22, "subtype": "inline"}, {"bbox": [186, 662, 215, 673], "content": "n\\neq4", "parent_index": 22, "subtype": "inline"}, {"bbox": [266, 660, 340, 673], "content": "\\rho(\\sigma_{i})=1+A_{i}", "parent_index": 22, "subtype": "inline"}, {"bbox": [405, 660, 450, 673], "content": ":(A_{i})=k", "parent_index": 22, "subtype": "inline"}, {"bbox": [168, 675, 329, 687], "content": "r=d i m V\\leq(n-1)(k-1)+1", "parent_index": 23, "subtype": "inline"}, {"bbox": [229, 690, 257, 699], "content": "k=2", "parent_index": 23, "subtype": "inline"}, {"bbox": [264, 689, 339, 699], "content": "r=d i m V=n", "parent_index": 23, "subtype": "inline"}, {"bbox": [378, 689, 418, 699], "content": "V=\\mathbb{C}^{n}", "parent_index": 23, "subtype": "inline"}]
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[]
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Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\leq j\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$ . Then
$$
\dim(I m(A_{i_{1}}))=k
$$
$$
\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\leq k+k-1=2k-1
$$
$$
\dim(I m(A_{i_{1}})+\cdot\cdot\cdot+I m(A_{i_{n-1}}))\leq k+(n-2)(k-1)=(n-1)(k-1)+1.
$$
Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following
Theorem 3.11. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $\cdot(A_{i})=2$ .
Then $r=n$ and one of the following holds.
(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ .
(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors.
# 4. For corank 2 the friendship graph is a chai
In this section, we assume throughout that we have an irreducible representation
$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$
where $r\geq n$ , and
$$
\rho(\sigma_{i})=1+A_{i},\;\;r a n k(A_{i})=2,\;\;1\leq i\leq n-1.
$$
Theorem 4.1. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\geq n$ and $n\geq6$ . Let $r a n k(A_{1})=2$ .
Then $I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$ for $1\,\leq\,i\,\leq\,n\,-\,2$ ; that is the friendship graph of $\rho$ contains the chain graph.
Proof. Suppose not. Then by Theorem $3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider
$$
U=I m(A_{1})+I m(A_{2})+I m(A_{3}).
$$
Since $I m(A_{1})\cap I m(A_{3})\neq0$ , $d i m U\leq5$ .
For $i=4,\dots,n-1$ , let $a_{i},\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\cap I m(A_{i})$ and $I m(A_{2})\cap I m(A_{i})$ . Since $I m(A_{1})\cap I m(A_{2})=0$ ,
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<html><body>
<p data-bbox="123 110 486 168">Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\leq j\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$ . Then </p>
<div class="equation" data-bbox="259 183 352 197">$$
\dim(I m(A_{i_{1}}))=k
$$</div>
<div class="equation" data-bbox="185 221 426 234">$$
\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\leq k+k-1=2k-1
$$</div>
<div class="equation" data-bbox="125 252 484 270">$$
\dim(I m(A_{i_{1}})+\cdot\cdot\cdot+I m(A_{i_{n-1}}))\leq k+(n-2)(k-1)=(n-1)(k-1)+1.
$$</div>
<p data-bbox="124 284 487 313">Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following </p>
<p data-bbox="124 319 486 348">Theorem 3.11. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $\cdot(A_{i})=2$ . </p>
<p data-bbox="137 348 358 361">Then $r=n$ and one of the following holds. </p>
<p data-bbox="125 362 487 389">(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . </p>
<p data-bbox="126 390 487 418">(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. </p>
<h1 data-bbox="157 429 444 443">4. For corank 2 the friendship graph is a chai </h1>
<p data-bbox="125 450 487 477">In this section, we assume throughout that we have an irreducible representation </p>
<div class="equation" data-bbox="259 480 351 494">$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$</div>
<p data-bbox="125 496 216 510">where $r\geq n$ , and </p>
<div class="equation" data-bbox="186 516 424 531">$$
\rho(\sigma_{i})=1+A_{i},\;\;r a n k(A_{i})=2,\;\;1\leq i\leq n-1.
$$</div>
<p data-bbox="124 541 486 569">Theorem 4.1. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\geq n$ and $n\geq6$ . Let $r a n k(A_{1})=2$ . </p>
<p data-bbox="124 570 486 598">Then $I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$ for $1\,\leq\,i\,\leq\,n\,-\,2$ ; that is the friendship graph of $\rho$ contains the chain graph. </p>
<p data-bbox="124 604 486 633">Proof. Suppose not. Then by Theorem $3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider </p>
<div class="equation" data-bbox="217 639 392 654">$$
U=I m(A_{1})+I m(A_{2})+I m(A_{3}).
$$</div>
<p data-bbox="124 657 332 672">Since $I m(A_{1})\cap I m(A_{3})\neq0$ , $d i m U\leq5$ . </p>
<p data-bbox="125 672 487 701">For $i=4,\dots,n-1$ , let $a_{i},\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\cap I m(A_{i})$ and $I m(A_{2})\cap I m(A_{i})$ . Since $I m(A_{1})\cap I m(A_{2})=0$ , </p>
</body></html>
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0003047v1
| 8 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\\leq j\\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}$ . Then ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}}))=k\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following ", "page_idx": 8}, {"type": "text", "text": "Theorem 3.11. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $\\cdot(A_{i})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $r=n$ and one of the following holds. ", "page_idx": 8}, {"type": "text", "text": "(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . ", "page_idx": 8}, {"type": "text", "text": "(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. ", "page_idx": 8}, {"type": "text", "text": "4. For corank 2 the friendship graph is a chai ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "In this section, we assume throughout that we have an irreducible representation ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "where $r\\geq n$ , and ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Theorem 4.1. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\geq n$ and $n\\geq6$ . Let $r a n k(A_{1})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}$ for $1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2$ ; that is the friendship graph of $\\rho$ contains the chain graph. ", "page_idx": 8}, {"type": "text", "text": "Proof. Suppose not. Then by Theorem $3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider ", "page_idx": 8}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+I m(A_{3}).\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Since $I m(A_{1})\\cap I m(A_{3})\\neq0$ , $d i m U\\leq5$ . ", "page_idx": 8}, {"type": "text", "text": "For $i=4,\\dots,n-1$ , let $a_{i},\\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\\cap I m(A_{i})$ and $I m(A_{2})\\cap I m(A_{i})$ . Since $I m(A_{1})\\cap I m(A_{2})=0$ , ", "page_idx": 8}]
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Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 89, 389, 101], "lines": [{"bbox": [223, 93, 388, 101], "spans": [{"bbox": [223, 93, 388, 101], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [478, 91, 486, 100], "lines": [{"bbox": [479, 93, 486, 102], "spans": [{"bbox": [479, 93, 486, 102], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 486, 168], "lines": [{"bbox": [137, 112, 485, 127], "spans": [{"bbox": [137, 112, 485, 127], "score": 1.0, "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 486, 141], "score": 1.0, "content": "tion is connected. Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [124, 112, 487, 171]}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [125, 286, 486, 316]}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 322, 486, 350]}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [140, 350, 357, 363]}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [126, 363, 486, 391]}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [126, 392, 487, 420]}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [125, 452, 486, 480]}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [124, 496, 217, 512]}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [126, 543, 485, 571]}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [126, 569, 487, 600]}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [125, 605, 487, 635]}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [126, 659, 331, 674]}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 673, 487, 702]}]}
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[{"type": "text", "bbox": [123, 110, 486, 168], "content": "Proof. By corollary 3.9, the friendship graph of the representa- tion is connected. Arrange the vertices of the graph in a sequence such that each term , , is a friend of one the terms . Then", "index": 0}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "content": "", "index": 1}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "content": "", "index": 2}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "content": "", "index": 3}, {"type": "text", "bbox": [124, 284, 487, 313], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following", "index": 4}, {"type": "text", "bbox": [124, 319, 486, 348], "content": "Theorem 3.11. Let be irreducible, where , . Suppose , where rank .", "index": 5}, {"type": "text", "bbox": [137, 348, 358, 361], "content": "Then and one of the following holds.", "index": 6}, {"type": "text", "bbox": [125, 362, 487, 389], "content": "(a) The full friendship graph has an edge between and for all .", "index": 7}, {"type": "text", "bbox": [126, 390, 487, 418], "content": "(b) The full friendship graph has an edge between and whenever and are not neighbors.", "index": 8}, {"type": "title", "bbox": [157, 429, 444, 443], "content": "4. For corank 2 the friendship graph is a chai", "index": 9}, {"type": "text", "bbox": [125, 450, 487, 477], "content": "In this section, we assume throughout that we have an irreducible representation", "index": 10}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "content": "", "index": 11}, {"type": "text", "bbox": [125, 496, 216, 510], "content": "where , and", "index": 12}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "content": "", "index": 13}, {"type": "text", "bbox": [124, 541, 486, 569], "content": "Theorem 4.1. Let be an irreducible representation, where and . Let .", "index": 14}, {"type": "text", "bbox": [124, 570, 486, 598], "content": "Then for ; that is the friendship graph of contains the chain graph.", "index": 15}, {"type": "text", "bbox": [124, 604, 486, 633], "content": "Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider", "index": 16}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "content": "", "index": 17}, {"type": "text", "bbox": [124, 657, 332, 672], "content": "Since , .", "index": 18}, {"type": "text", "bbox": [125, 672, 487, 701], "content": "For , let be, respectively, nonzero elements of and . Since ,", "index": 19}]
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[{"bbox": [137, 112, 485, 127], "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "parent_index": 0, "line_index": 0}, {"bbox": [125, 127, 486, 141], "content": "tion is connected. Arrange the vertices of the graph in a sequence", "parent_index": 0, "line_index": 1}, {"bbox": [126, 139, 487, 157], "content": "such that each term , , is a friend", "parent_index": 0, "line_index": 2}, {"bbox": [124, 154, 339, 171], "content": "of one the terms . Then", "parent_index": 0, "line_index": 3}, {"bbox": [137, 286, 486, 300], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "parent_index": 4, "line_index": 0}, {"bbox": [125, 299, 174, 316], "content": "following", "parent_index": 4, "line_index": 1}, {"bbox": [125, 322, 486, 336], "content": "Theorem 3.11. Let be irreducible, where", "parent_index": 5, "line_index": 0}, {"bbox": [126, 336, 453, 350], "content": ", . Suppose , where rank .", "parent_index": 5, "line_index": 1}, {"bbox": [140, 350, 357, 363], "content": "Then and one of the following holds.", "parent_index": 6, "line_index": 0}, {"bbox": [139, 363, 486, 378], "content": "(a) The full friendship graph has an edge between and for", "parent_index": 7, "line_index": 0}, {"bbox": [126, 377, 152, 391], "content": "all .", "parent_index": 7, "line_index": 1}, {"bbox": [139, 392, 487, 406], "content": "(b) The full friendship graph has an edge between and whenever", "parent_index": 8, "line_index": 0}, {"bbox": [126, 405, 275, 420], "content": "and are not neighbors.", "parent_index": 8, "line_index": 1}, {"bbox": [156, 431, 445, 444], "content": "4. For corank 2 the friendship graph is a chai", "parent_index": 9, "line_index": 0}, {"bbox": [137, 452, 486, 465], "content": "In this section, we assume throughout that we have an irreducible", "parent_index": 10, "line_index": 0}, {"bbox": [125, 466, 200, 480], "content": "representation", "parent_index": 10, "line_index": 1}, {"bbox": [124, 496, 217, 512], "content": "where , and", "parent_index": 12, "line_index": 0}, {"bbox": [126, 543, 485, 558], "content": "Theorem 4.1. Let be an irreducible representation,", "parent_index": 14, "line_index": 0}, {"bbox": [127, 558, 344, 571], "content": "where and . Let .", "parent_index": 14, "line_index": 1}, {"bbox": [137, 569, 487, 587], "content": "Then for ; that is the", "parent_index": 15, "line_index": 0}, {"bbox": [126, 585, 365, 600], "content": "friendship graph of contains the chain graph.", "parent_index": 15, "line_index": 1}, {"bbox": [136, 605, 487, 622], "content": "Proof. Suppose not. Then by Theorem", "parent_index": 16, "line_index": 0}, {"bbox": [125, 621, 386, 635], "content": "0 whenever and are not neighbors. Consider", "parent_index": 16, "line_index": 1}, {"bbox": [126, 659, 331, 674], "content": "Since , .", "parent_index": 18, "line_index": 0}, {"bbox": [137, 673, 487, 688], "content": "For , let be, respectively, nonzero elements of", "parent_index": 19, "line_index": 0}, {"bbox": [126, 687, 484, 702], "content": "and . Since ,", "parent_index": 19, "line_index": 1}]
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[]
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[{"bbox": [126, 142, 216, 155], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [327, 142, 343, 156], "content": "A_{i_{j}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [353, 142, 426, 154], "content": "2\\leq j\\leq n-1", "parent_index": 0, "subtype": "inline"}, {"bbox": [213, 156, 304, 169], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [259, 183, 352, 197], "content": "\\dim(I m(A_{i_{1}}))=k", "parent_index": 1, "subtype": "interline"}, {"bbox": [185, 221, 426, 234], "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "parent_index": 2, "subtype": "interline"}, {"bbox": [125, 252, 484, 270], "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "parent_index": 3, "subtype": "interline"}, {"bbox": [238, 323, 342, 335], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "parent_index": 5, "subtype": "inline"}, {"bbox": [462, 323, 486, 335], "content": "r=", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 338, 179, 348], "content": "d i m V\\geq n", "parent_index": 5, "subtype": "inline"}, {"bbox": [186, 338, 215, 349], "content": "n\\neq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [266, 337, 340, 350], "content": "\\rho(\\sigma_{i})=1+A_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [405, 336, 449, 349], "content": "\\cdot(A_{i})=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [168, 354, 197, 360], "content": "r=n", "parent_index": 6, "subtype": "inline"}, {"bbox": [401, 364, 414, 376], "content": "A_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [442, 364, 466, 377], "content": "A_{i+1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [142, 380, 147, 388], "content": "i", "parent_index": 7, "subtype": "inline"}, {"bbox": [384, 392, 397, 404], "content": "A_{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [421, 392, 435, 406], "content": "A_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 407, 138, 418], "content": "A_{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [165, 407, 178, 420], "content": "A_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [259, 480, 351, 494], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "parent_index": 11, "subtype": "interline"}, {"bbox": [159, 500, 189, 510], "content": "r\\geq n", "parent_index": 12, "subtype": "inline"}, {"bbox": [186, 516, 424, 531], "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "parent_index": 13, "subtype": "interline"}, {"bbox": [229, 543, 318, 557], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 14, "subtype": "inline"}, {"bbox": [159, 559, 188, 570], "content": "r\\geq n", "parent_index": 14, "subtype": "inline"}, {"bbox": [214, 559, 244, 570], "content": "n\\geq6", "parent_index": 14, "subtype": "inline"}, {"bbox": [271, 558, 341, 571], "content": "r a n k(A_{1})=2", "parent_index": 14, "subtype": "inline"}, {"bbox": [169, 571, 312, 585], "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "parent_index": 15, "subtype": "inline"}, {"bbox": [336, 572, 421, 584], "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "parent_index": 15, "subtype": "inline"}, {"bbox": [226, 588, 233, 599], "content": "\\rho", "parent_index": 15, "subtype": "inline"}, {"bbox": [347, 606, 487, 621], "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "parent_index": 16, "subtype": "inline"}, {"bbox": [187, 623, 199, 633], "content": "A_{i}", "parent_index": 16, "subtype": "inline"}, {"bbox": [226, 622, 239, 635], "content": "A_{j}", "parent_index": 16, "subtype": "inline"}, {"bbox": [217, 639, 392, 654], "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "parent_index": 17, "subtype": "interline"}, {"bbox": [156, 660, 270, 673], "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "parent_index": 18, "subtype": "inline"}, {"bbox": [276, 660, 328, 672], "content": "d i m U\\leq5", "parent_index": 18, "subtype": "inline"}, {"bbox": [159, 676, 240, 687], "content": "i=4,\\dots,n-1", "parent_index": 19, "subtype": "inline"}, {"bbox": [264, 674, 293, 686], "content": "a_{i},\\ b_{i}", "parent_index": 19, "subtype": "inline"}, {"bbox": [126, 689, 216, 702], "content": "I m(A_{1})\\cap I m(A_{i})", "parent_index": 19, "subtype": "inline"}, {"bbox": [242, 688, 332, 702], "content": "I m(A_{2})\\cap I m(A_{i})", "parent_index": 19, "subtype": "inline"}, {"bbox": [369, 689, 482, 702], "content": "I m(A_{1})\\cap I m(A_{2})=0", "parent_index": 19, "subtype": "inline"}]
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[]
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$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$ . Thus
$$
U=I m(A_{1})+I m(A_{2})+\cdot\cdot\cdot+I m(A_{n-1}),
$$
which is invariant under $\rho(B_{n})$ . Thus $r\leq5$ , by the irreducibility of $\rho$ , a contradiction with $r\geq n\geq6$ .
Remark 4.2. For $n\,=\,5$ and $\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:
By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296).
Lemma 4.3. Let $\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\,\geq\,n$ , $n\,\geq\,5$ , and $r a n k(A_{1})\,=\,2$ . Suppose that the associated friendship graph contains the chain.
Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors).
Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\rho$ . Then
$$
I m(A_{i})\cap I m(A_{i+1})\neq\{0\}
$$
for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence $I m(A_{i})\cap I m(A_{i+1})$ are one-dimensional.
For any $x\in I m(A_{i})$ , $x=A_{i}y$ , $x\neq0$ , we have that
$$
T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\in I m(A_{i+1})
$$
for $T=\rho(\tau)$ . Moreover, $T x\neq0$ because $T$ is invertible.
Choose $x_{1}~\neq~0$ to be a basis vector for $I m(A_{1})\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\leq i\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ .
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<html><body>
<p data-bbox="124 110 486 139">$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$ . Thus </p>
<div class="equation" data-bbox="198 147 411 161">$$
U=I m(A_{1})+I m(A_{2})+\cdot\cdot\cdot+I m(A_{n-1}),
$$</div>
<p data-bbox="124 165 486 194">which is invariant under $\rho(B_{n})$ . Thus $r\leq5$ , by the irreducibility of $\rho$ , a contradiction with $r\geq n\geq6$ . </p>
<p data-bbox="124 200 486 243">Remark 4.2. For $n\,=\,5$ and $\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: </p>
<div class="image" data-bbox="124 270 252 335"><img data-bbox="124 270 252 335"/></div>
<p data-bbox="124 365 486 408">By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296). </p>
<p data-bbox="124 421 486 464">Lemma 4.3. Let $\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\,\geq\,n$ , $n\,\geq\,5$ , and $r a n k(A_{1})\,=\,2$ . Suppose that the associated friendship graph contains the chain. </p>
<p data-bbox="124 465 485 493">Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). </p>
<p data-bbox="124 500 486 528">Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\rho$ . Then </p>
<div class="equation" data-bbox="239 532 372 545">$$
I m(A_{i})\cap I m(A_{i+1})\neq\{0\}
$$</div>
<p data-bbox="124 547 486 603">for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence $I m(A_{i})\cap I m(A_{i+1})$ are one-dimensional. </p>
<p data-bbox="138 604 398 618">For any $x\in I m(A_{i})$ , $x=A_{i}y$ , $x\neq0$ , we have that </p>
<div class="equation" data-bbox="173 625 437 639">$$
T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\in I m(A_{i+1})
$$</div>
<p data-bbox="124 643 412 658">for $T=\rho(\tau)$ . Moreover, $T x\neq0$ because $T$ is invertible. </p>
<p data-bbox="124 658 487 701">Choose $x_{1}~\neq~0$ to be a basis vector for $I m(A_{1})\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\leq i\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ . </p>
</body></html>
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0003047v1
| 9 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})$ . Thus ", "page_idx": 9}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "which is invariant under $\\rho(B_{n})$ . Thus $r\\leq5$ , by the irreducibility of $\\rho$ , a contradiction with $r\\geq n\\geq6$ . ", "page_idx": 9}, {"type": "text", "text": "Remark 4.2. For $n\\,=\\,5$ and $\\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: ", "page_idx": 9}, {"type": "image", "img_path": "images/37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296). ", "page_idx": 9}, {"type": "text", "text": "Lemma 4.3. Let $\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\,\\geq\\,n$ , $n\\,\\geq\\,5$ , and $r a n k(A_{1})\\,=\\,2$ . Suppose that the associated friendship graph contains the chain. ", "page_idx": 9}, {"type": "text", "text": "Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). ", "page_idx": 9}, {"type": "text", "text": "Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\\rho$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence $I m(A_{i})\\cap I m(A_{i+1})$ are one-dimensional. ", "page_idx": 9}, {"type": "text", "text": "For any $x\\in I m(A_{i})$ , $x=A_{i}y$ , $x\\neq0$ , we have that ", "page_idx": 9}, {"type": "equation", "text": "$$\nT x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for $T=\\rho(\\tau)$ . Moreover, $T x\\neq0$ because $T$ is invertible. ", "page_idx": 9}, {"type": "text", "text": "Choose $x_{1}~\\neq~0$ to be a basis vector for $I m(A_{1})\\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\\leq i\\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ . ", "page_idx": 9}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 110, 486, 139], "lines": [{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". Thus", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [124, 165, 486, 194], "lines": [{"bbox": [125, 166, 485, 182], "spans": [{"bbox": [125, 166, 253, 182], "score": 1.0, "content": "which is invariant under ", "type": "text"}, {"bbox": [254, 168, 284, 181], "score": 0.94, "content": "\\rho(B_{n})", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [284, 166, 320, 182], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones’ representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 91, 137, 100], "lines": [{"bbox": [125, 93, 137, 103], "spans": [{"bbox": [125, 93, 137, 103], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 486, 139], "lines": [{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". Thus", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 114, 485, 141]}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [124, 165, 486, 194], "lines": [{"bbox": [125, 166, 485, 182], "spans": [{"bbox": [125, 166, 253, 182], "score": 1.0, "content": "which is invariant under ", "type": "text"}, {"bbox": [254, 168, 284, 181], "score": 0.94, "content": "\\rho(B_{n})", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [284, 166, 320, 182], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [125, 166, 485, 195]}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6, "bbox_fs": [125, 203, 486, 244]}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones’ representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [125, 368, 486, 409]}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [125, 424, 487, 465]}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [126, 465, 485, 495]}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [126, 502, 485, 530]}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [125, 550, 486, 605]}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [137, 604, 398, 620]}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [126, 645, 411, 660]}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [126, 659, 486, 703]}]}
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[{"type": "text", "bbox": [124, 110, 486, 139], "content": "and are linearly independent, so they are a basis for , and . Thus", "index": 0}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "content": "", "index": 1}, {"type": "text", "bbox": [124, 165, 486, 194], "content": "which is invariant under . Thus , by the irreducibility of , a contradiction with .", "index": 2}, {"type": "text", "bbox": [124, 200, 486, 243], "content": "Remark 4.2. For and satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:", "index": 3}, {"type": "image", "bbox": [124, 270, 252, 335], "content": "", "index": 4}, {"type": "text", "bbox": [124, 365, 486, 408], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to of the Jones’ representation (see [3], p. 296).", "index": 5}, {"type": "text", "bbox": [124, 421, 486, 464], "content": "Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.", "index": 6}, {"type": "text", "bbox": [124, 465, 485, 493], "content": "Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).", "index": 7}, {"type": "text", "bbox": [124, 500, 486, 528], "content": "Proof. By corollary 3.10, . Consider the full friendship graph of . Then", "index": 8}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "content": "", "index": 9}, {"type": "text", "bbox": [124, 547, 486, 603], "content": "for any where indices are taken modulo . If is two-dimensional, then , and is a two- dimensional invariant subspace, contradicting the irreducibility of . Hence are one-dimensional.", "index": 10}, {"type": "text", "bbox": [138, 604, 398, 618], "content": "For any , , , we have that", "index": 11}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "content": "", "index": 12}, {"type": "text", "bbox": [124, 643, 412, 658], "content": "for . Moreover, because is invertible.", "index": 13}, {"type": "text", "bbox": [124, 658, 487, 701], "content": "Choose to be a basis vector for . Define for . Then is a basis vector for .", "index": 14}]
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[{"bbox": [126, 114, 485, 127], "content": "and are linearly independent, so they are a basis for , and", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 307, 141], "content": ". Thus", "parent_index": 0, "line_index": 1}, {"bbox": [125, 166, 485, 182], "content": "which is invariant under . Thus , by the irreducibility of ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 182, 288, 195], "content": "a contradiction with .", "parent_index": 2, "line_index": 1}, {"bbox": [125, 203, 486, 217], "content": "Remark 4.2. For and satisfying the hypothesis of theorem", "parent_index": 3, "line_index": 0}, {"bbox": [126, 217, 486, 231], "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "parent_index": 3, "line_index": 1}, {"bbox": [126, 231, 266, 244], "content": "and 2) an exceptional case:", "parent_index": 3, "line_index": 2}, {"bbox": [137, 368, 485, 381], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "parent_index": 5, "line_index": 0}, {"bbox": [125, 381, 486, 396], "content": "the above friendship graph is equivalent to the restriction to of the", "parent_index": 5, "line_index": 1}, {"bbox": [126, 396, 322, 409], "content": "Jones’ representation (see [3], p. 296).", "parent_index": 5, "line_index": 2}, {"bbox": [125, 424, 485, 439], "content": "Lemma 4.3. Let be an irreducible representation,", "parent_index": 6, "line_index": 0}, {"bbox": [127, 439, 487, 453], "content": "where , , and . Suppose that the associated", "parent_index": 6, "line_index": 1}, {"bbox": [127, 454, 310, 465], "content": "friendship graph contains the chain.", "parent_index": 6, "line_index": 2}, {"bbox": [138, 465, 485, 482], "content": "Then and the associated friendship graph is the chain (that is,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 480, 321, 495], "content": "the only edges are between neighbors).", "parent_index": 7, "line_index": 1}, {"bbox": [137, 502, 485, 516], "content": "Proof. By corollary 3.10, . Consider the full friendship graph", "parent_index": 8, "line_index": 0}, {"bbox": [126, 516, 181, 530], "content": "of . Then", "parent_index": 8, "line_index": 1}, {"bbox": [126, 550, 486, 564], "content": "for any where indices are taken modulo . If is", "parent_index": 10, "line_index": 0}, {"bbox": [125, 563, 484, 577], "content": "two-dimensional, then , and is a two-", "parent_index": 10, "line_index": 1}, {"bbox": [126, 577, 485, 591], "content": "dimensional invariant subspace, contradicting the irreducibility of .", "parent_index": 10, "line_index": 2}, {"bbox": [126, 591, 370, 605], "content": "Hence are one-dimensional.", "parent_index": 10, "line_index": 3}, {"bbox": [137, 604, 398, 620], "content": "For any , , , we have that", "parent_index": 11, "line_index": 0}, {"bbox": [126, 645, 411, 660], "content": "for . Moreover, because is invertible.", "parent_index": 13, "line_index": 0}, {"bbox": [138, 659, 486, 674], "content": "Choose to be a basis vector for . Define", "parent_index": 14, "line_index": 0}, {"bbox": [126, 674, 485, 688], "content": "for . Then is a basis vector for", "parent_index": 14, "line_index": 1}, {"bbox": [126, 686, 179, 703], "content": ".", "parent_index": 14, "line_index": 2}]
|
[{"bbox": [124, 270, 252, 335], "content": "", "parent_index": 4, "subtype": "body"}]
|
[{"bbox": [126, 118, 136, 125], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [162, 115, 171, 125], "content": "b_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [421, 114, 459, 127], "content": "I m(A_{i})", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 272, 140], "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "parent_index": 0, "subtype": "inline"}, {"bbox": [198, 147, 411, 161], "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "parent_index": 1, "subtype": "interline"}, {"bbox": [254, 168, 284, 181], "content": "\\rho(B_{n})", "parent_index": 2, "subtype": "inline"}, {"bbox": [320, 169, 349, 180], "content": "r\\leq5", "parent_index": 2, "subtype": "inline"}, {"bbox": [475, 172, 482, 180], "content": "\\rho", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 184, 284, 194], "content": "r\\geq n\\geq6", "parent_index": 2, "subtype": "inline"}, {"bbox": [225, 205, 257, 213], "content": "n\\,=\\,5", "parent_index": 3, "subtype": "inline"}, {"bbox": [285, 208, 292, 216], "content": "\\rho", "parent_index": 3, "subtype": "inline"}, {"bbox": [438, 383, 451, 394], "content": "B_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [221, 426, 314, 438], "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "parent_index": 6, "subtype": "inline"}, {"bbox": [160, 441, 191, 451], "content": "r\\,\\geq\\,n", "parent_index": 6, "subtype": "inline"}, {"bbox": [199, 441, 231, 451], "content": "n\\,\\geq\\,5", "parent_index": 6, "subtype": "inline"}, {"bbox": [265, 440, 336, 452], "content": "r a n k(A_{1})\\,=\\,2", "parent_index": 6, "subtype": "inline"}, {"bbox": [168, 471, 196, 477], "content": "r=n", "parent_index": 7, "subtype": "inline"}, {"bbox": [275, 507, 304, 513], "content": "r=n", "parent_index": 8, "subtype": "inline"}, {"bbox": [139, 522, 146, 529], "content": "\\rho", "parent_index": 8, "subtype": "inline"}, {"bbox": [239, 532, 372, 545], "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "parent_index": 9, "subtype": "interline"}, {"bbox": [167, 552, 171, 560], "content": "i", "parent_index": 10, "subtype": "inline"}, {"bbox": [345, 555, 352, 560], "content": "n", "parent_index": 10, "subtype": "inline"}, {"bbox": [372, 551, 473, 563], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [243, 564, 369, 577], "content": "I m(A_{1})=I m(A_{2})=\\ldots", "parent_index": 10, "subtype": "inline"}, {"bbox": [398, 564, 437, 577], "content": "I m(A_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [475, 582, 482, 590], "content": "\\rho", "parent_index": 10, "subtype": "inline"}, {"bbox": [161, 592, 260, 605], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [181, 606, 240, 619], "content": "x\\in I m(A_{i})", "parent_index": 11, "subtype": "inline"}, {"bbox": [247, 607, 288, 618], "content": "x=A_{i}y", "parent_index": 11, "subtype": "inline"}, {"bbox": [294, 607, 324, 618], "content": "x\\neq0", "parent_index": 11, "subtype": "inline"}, {"bbox": [173, 625, 437, 639], "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "parent_index": 12, "subtype": "interline"}, {"bbox": [144, 647, 190, 659], "content": "T=\\rho(\\tau)", "parent_index": 13, "subtype": "inline"}, {"bbox": [252, 648, 288, 659], "content": "T x\\neq0", "parent_index": 13, "subtype": "inline"}, {"bbox": [335, 648, 344, 656], "content": "T", "parent_index": 13, "subtype": "inline"}, {"bbox": [180, 662, 217, 673], "content": "x_{1}~\\neq~0", "parent_index": 14, "subtype": "inline"}, {"bbox": [351, 661, 444, 673], "content": "I m(A_{1})\\cap I m(A_{2})", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 674, 187, 687], "content": "x_{i+1}=T^{i}x_{1}", "parent_index": 14, "subtype": "inline"}, {"bbox": [208, 676, 280, 686], "content": "1\\leq i\\leq n-1", "parent_index": 14, "subtype": "inline"}, {"bbox": [318, 679, 328, 686], "content": "x_{i}", "parent_index": 14, "subtype": "inline"}, {"bbox": [436, 675, 485, 687], "content": "I m(A_{i})\\cap", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 689, 174, 702], "content": "I m(A_{i+1})", "parent_index": 14, "subtype": "inline"}]
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[]
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If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then
$$
A_{i}A_{j}=A_{j}A_{i}
$$
and
$$
A_{i}A_{j+1}=A_{j+1}A_{i}.
$$
So, if $x\in I m(A_{j})\cap I m(A_{j+1})$ then $A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$ . But this means that $s p a n\{x_{1}\}$ is an invariant subspace and the representation is not irreducible.
So, if the representation is irreducible, then for any $i$ , $x_{i}\notin s p a n\{x_{i+1}\}$ . From this follows that for any $i$
$$
I m(A_{i})=s p a n\{x_{i-1},x_{i}\}
$$
and the $n$ vectors $x_{0},x_{1},\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$
$$
I m(A_{i})\cap I m(A_{j})=\{0\}.
$$
Now, we have the following
Theorem 4.4. Let $\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$ be irreducible, where $r\ \geq\ n$ .
Suppose that for any generator $\sigma_{i}$ , $\rho(\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ .
1) If $n\,\geq\,6$ , then $r\,=\,n$ and $\rho$ has a friendship graph which is a chain.
2) If $n=5$ , then $r=5$ and either $\rho$ has a friendship graph which is a chain or $\rho$ has the exceptional friendship graph (see Remark 4.2).
3) If $n=4$ , then either $r=4$ and $\rho$ has a friendship graph which is a chain; or $\rho$ has one of the following exceptional friendship graphs:
Proof. 1) If $n\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ .
2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma
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<html><body>
<p data-bbox="124 110 487 167">If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then </p>
<div class="equation" data-bbox="271 177 339 190">$$
A_{i}A_{j}=A_{j}A_{i}
$$</div>
<p data-bbox="125 194 147 208">and </p>
<div class="equation" data-bbox="259 214 351 227">$$
A_{i}A_{j+1}=A_{j+1}A_{i}.
$$</div>
<p data-bbox="124 229 487 270">So, if $x\in I m(A_{j})\cap I m(A_{j+1})$ then $A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$ . But this means that $s p a n\{x_{1}\}$ is an invariant subspace and the representation is not irreducible. </p>
<p data-bbox="124 271 488 299">So, if the representation is irreducible, then for any $i$ , $x_{i}\notin s p a n\{x_{i+1}\}$ . From this follows that for any $i$ </p>
<div class="equation" data-bbox="242 308 369 322">$$
I m(A_{i})=s p a n\{x_{i-1},x_{i}\}
$$</div>
<p data-bbox="124 327 486 356">and the $n$ vectors $x_{0},x_{1},\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$ </p>
<div class="equation" data-bbox="242 365 368 378">$$
I m(A_{i})\cap I m(A_{j})=\{0\}.
$$</div>
<p data-bbox="136 383 280 397">Now, we have the following </p>
<p data-bbox="124 405 486 433">Theorem 4.4. Let $\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$ be irreducible, where $r\ \geq\ n$ .
Suppose that for any generator $\sigma_{i}$ , $\rho(\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ . </p>
<p data-bbox="125 434 486 460">1) If $n\,\geq\,6$ , then $r\,=\,n$ and $\rho$ has a friendship graph which is a chain. </p>
<p data-bbox="126 461 487 489">2) If $n=5$ , then $r=5$ and either $\rho$ has a friendship graph which is a chain or $\rho$ has the exceptional friendship graph (see Remark 4.2). </p>
<p data-bbox="126 489 487 518">3) If $n=4$ , then either $r=4$ and $\rho$ has a friendship graph which is a chain; or $\rho$ has one of the following exceptional friendship graphs: </p>
<div class="image" data-bbox="122 541 464 602"><img data-bbox="122 541 464 602"/></div>
<p data-bbox="123 630 486 671">Proof. 1) If $n\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ . </p>
<p data-bbox="124 672 487 700">2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma </p>
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0003047v1
| 10 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j}=A_{j}A_{i}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j+1}=A_{j+1}A_{i}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "So, if $x\\in I m(A_{j})\\cap I m(A_{j+1})$ then $A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})$ . But this means that $s p a n\\{x_{1}\\}$ is an invariant subspace and the representation is not irreducible. ", "page_idx": 10}, {"type": "text", "text": "So, if the representation is irreducible, then for any $i$ , $x_{i}\\notin s p a n\\{x_{i+1}\\}$ . From this follows that for any $i$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and the $n$ vectors $x_{0},x_{1},\\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{j})=\\{0\\}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now, we have the following ", "page_idx": 10}, {"type": "text", "text": "Theorem 4.4. Let $\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})$ be irreducible, where $r\\ \\geq\\ n$ . \nSuppose that for any generator $\\sigma_{i}$ , $\\rho(\\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ . ", "page_idx": 10}, {"type": "text", "text": "1) If $n\\,\\geq\\,6$ , then $r\\,=\\,n$ and $\\rho$ has a friendship graph which is a chain. ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then $r=5$ and either $\\rho$ has a friendship graph which is a chain or $\\rho$ has the exceptional friendship graph (see Remark 4.2). ", "page_idx": 10}, {"type": "text", "text": "3) If $n=4$ , then either $r=4$ and $\\rho$ has a friendship graph which is a chain; or $\\rho$ has one of the following exceptional friendship graphs: ", "page_idx": 10}, {"type": "image", "img_path": "images/27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 10}, {"type": "text", "text": "Proof. 1) If $n\\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ . ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma ", "page_idx": 10}]
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Then, because", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 443, 155], "score": 1.0, "content": "we have 5 or more vertices in the full friendship graph, for any ", "type": "text"}, {"bbox": [443, 142, 455, 153], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [456, 141, 486, 155], "score": 1.0, "content": " there", "type": "text"}], "index": 2}, {"bbox": [126, 155, 469, 169], "spans": [{"bbox": [126, 155, 158, 169], "score": 1.0, "content": "exists ", "type": "text"}, {"bbox": [158, 157, 164, 168], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [164, 155, 246, 169], "score": 1.0, "content": " such that both ", "type": "text"}, {"bbox": [247, 156, 261, 169], "score": 0.93, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [261, 155, 286, 169], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [287, 156, 311, 169], "score": 0.93, "content": "A_{j+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [311, 155, 421, 169], "score": 1.0, "content": " are not neighbors of ", "type": "text"}, {"bbox": [421, 156, 434, 167], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [434, 155, 469, 169], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "lines": [{"bbox": [271, 177, 339, 190], "spans": [{"bbox": [271, 177, 339, 190], "score": 0.92, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 194, 147, 208], "lines": [{"bbox": [125, 197, 147, 209], "spans": [{"bbox": [125, 197, 147, 209], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "lines": [{"bbox": [259, 214, 351, 227], "spans": [{"bbox": [259, 214, 351, 227], "score": 0.9, "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 229, 487, 270], "lines": [{"bbox": [125, 231, 487, 246], "spans": [{"bbox": [125, 231, 154, 246], "score": 1.0, "content": "So, if", "type": "text"}, {"bbox": [155, 232, 275, 245], "score": 0.95, "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 120}, {"bbox": [276, 231, 304, 246], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [304, 232, 438, 245], "score": 0.95, "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [438, 231, 487, 246], "score": 1.0, "content": ". But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". 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Then, because", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 443, 155], "score": 1.0, "content": "we have 5 or more vertices in the full friendship graph, for any ", "type": "text"}, {"bbox": [443, 142, 455, 153], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [456, 141, 486, 155], "score": 1.0, "content": " there", "type": "text"}], "index": 2}, {"bbox": [126, 155, 469, 169], "spans": [{"bbox": [126, 155, 158, 169], "score": 1.0, "content": "exists ", "type": "text"}, {"bbox": [158, 157, 164, 168], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [164, 155, 246, 169], "score": 1.0, "content": " such that both ", "type": "text"}, {"bbox": [247, 156, 261, 169], "score": 0.93, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [261, 155, 286, 169], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [287, 156, 311, 169], "score": 0.93, "content": "A_{j+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [311, 155, 421, 169], "score": 1.0, "content": " are not neighbors of ", "type": "text"}, {"bbox": [421, 156, 434, 167], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [434, 155, 469, 169], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [125, 111, 486, 169]}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "lines": [{"bbox": [271, 177, 339, 190], "spans": [{"bbox": [271, 177, 339, 190], "score": 0.92, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 194, 147, 208], "lines": [{"bbox": [125, 197, 147, 209], "spans": [{"bbox": [125, 197, 147, 209], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [125, 197, 147, 209]}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "lines": [{"bbox": [259, 214, 351, 227], "spans": [{"bbox": [259, 214, 351, 227], "score": 0.9, "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 229, 487, 270], "lines": [{"bbox": [125, 231, 487, 246], "spans": [{"bbox": [125, 231, 154, 246], "score": 1.0, "content": "So, if", "type": "text"}, {"bbox": [155, 232, 275, 245], "score": 0.95, "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 120}, {"bbox": [276, 231, 304, 246], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [304, 232, 438, 245], "score": 0.95, "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [438, 231, 487, 246], "score": 1.0, "content": ". But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8, "bbox_fs": [125, 231, 487, 272]}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5, "bbox_fs": [125, 272, 489, 300]}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5, "bbox_fs": [125, 329, 486, 357]}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16, "bbox_fs": [137, 384, 279, 399]}, {"type": "list", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17, "is_list_end_line": true}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18, "is_list_start_line": true, "is_list_end_line": true}], "index": 17.5, "bbox_fs": [126, 407, 486, 435]}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 436, 487, 463]}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5, "bbox_fs": [127, 464, 487, 491]}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [127, 491, 486, 519]}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [124, 632, 486, 673]}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". If it contains a chain graph, then, by lemma", "type": "text"}], "index": 32}], "index": 31.5, "bbox_fs": [126, 674, 487, 702]}]}
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[{"type": "text", "bbox": [124, 110, 487, 167], "content": "If for some , is proportional to then, because a full friendship graph is a -graph, all the are proportional to . Then, because we have 5 or more vertices in the full friendship graph, for any there exists such that both and are not neighbors of . Then", "index": 0}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "content": "", "index": 1}, {"type": "text", "bbox": [125, 194, 147, 208], "content": "and", "index": 2}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "content": "", "index": 3}, {"type": "text", "bbox": [124, 229, 487, 270], "content": "So, if then . But this means that is an invariant subspace and the representation is not irreducible.", "index": 4}, {"type": "text", "bbox": [124, 271, 488, 299], "content": "So, if the representation is irreducible, then for any , . From this follows that for any", "index": 5}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "content": "", "index": 6}, {"type": "text", "bbox": [124, 327, 486, 356], "content": "and the vectors form a basis of . Then for any two non-neighbors and", "index": 7}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "content": "", "index": 8}, {"type": "text", "bbox": [136, 383, 280, 397], "content": "Now, we have the following", "index": 9}, {"type": "list", "bbox": [124, 405, 486, 433], "content": "", "index": 10}, {"type": "text", "bbox": [125, 434, 486, 460], "content": "1) If , then and has a friendship graph which is a chain.", "index": 11}, {"type": "text", "bbox": [126, 461, 487, 489], "content": "2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).", "index": 12}, {"type": "text", "bbox": [126, 489, 487, 518], "content": "3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:", "index": 13}, {"type": "image", "bbox": [122, 541, 464, 602], "content": "", "index": 14}, {"type": "text", "bbox": [123, 630, 486, 671], "content": "Proof. 1) If , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and .", "index": 15}, {"type": "text", "bbox": [124, 672, 487, 700], "content": "2) If , then by corollaries 3.9 and 3.10 the friendship graph of is connected and . If it contains a chain graph, then, by lemma", "index": 16}]
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[{"bbox": [136, 111, 485, 128], "content": "If for some , is proportional to then, because a full friendship", "parent_index": 0, "line_index": 0}, {"bbox": [125, 127, 486, 141], "content": "graph is a -graph, all the are proportional to . Then, because", "parent_index": 0, "line_index": 1}, {"bbox": [126, 141, 486, 155], "content": "we have 5 or more vertices in the full friendship graph, for any there", "parent_index": 0, "line_index": 2}, {"bbox": [126, 155, 469, 169], "content": "exists such that both and are not neighbors of . Then", "parent_index": 0, "line_index": 3}, {"bbox": [125, 197, 147, 209], "content": "and", "parent_index": 2, "line_index": 0}, {"bbox": [125, 231, 487, 246], "content": "So, if then . But this", "parent_index": 4, "line_index": 0}, {"bbox": [125, 246, 485, 259], "content": "means that is an invariant subspace and the representation", "parent_index": 4, "line_index": 1}, {"bbox": [125, 260, 216, 272], "content": "is not irreducible.", "parent_index": 4, "line_index": 2}, {"bbox": [137, 272, 489, 288], "content": "So, if the representation is irreducible, then for any , .", "parent_index": 5, "line_index": 0}, {"bbox": [125, 287, 287, 300], "content": "From this follows that for any", "parent_index": 5, "line_index": 1}, {"bbox": [125, 329, 486, 344], "content": "and the vectors form a basis of . Then for any two", "parent_index": 7, "line_index": 0}, {"bbox": [125, 342, 253, 357], "content": "non-neighbors and", "parent_index": 7, "line_index": 1}, {"bbox": [137, 384, 279, 399], "content": "Now, we have the following", "parent_index": 9, "line_index": 0}, {"bbox": [126, 407, 486, 421], "content": "Theorem 4.4. Let be irreducible, where .", "parent_index": 10, "line_index": 0}, {"bbox": [126, 421, 485, 435], "content": "Suppose that for any generator , , where .", "parent_index": 10, "line_index": 1}, {"bbox": [139, 436, 487, 449], "content": "1) If , then and has a friendship graph which is a", "parent_index": 11, "line_index": 0}, {"bbox": [126, 448, 158, 463], "content": "chain.", "parent_index": 11, "line_index": 1}, {"bbox": [138, 464, 487, 477], "content": "2) If , then and either has a friendship graph which is", "parent_index": 12, "line_index": 0}, {"bbox": [127, 477, 471, 491], "content": "a chain or has the exceptional friendship graph (see Remark 4.2).", "parent_index": 12, "line_index": 1}, {"bbox": [138, 491, 486, 504], "content": "3) If , then either and has a friendship graph which is", "parent_index": 13, "line_index": 0}, {"bbox": [127, 505, 474, 519], "content": "a chain; or has one of the following exceptional friendship graphs:", "parent_index": 13, "line_index": 1}, {"bbox": [137, 632, 485, 646], "content": "Proof. 1) If , then by theorem 4.1 the associated friendship", "parent_index": 15, "line_index": 0}, {"bbox": [124, 646, 486, 659], "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "parent_index": 15, "line_index": 1}, {"bbox": [126, 662, 158, 673], "content": ".", "parent_index": 15, "line_index": 2}, {"bbox": [137, 674, 487, 688], "content": "2) If , then by corollaries 3.9 and 3.10 the friendship graph of", "parent_index": 16, "line_index": 0}, {"bbox": [126, 688, 486, 702], "content": "is connected and . If it contains a chain graph, then, by lemma", "parent_index": 16, "line_index": 1}]
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4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.
3) If $n=4$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $\mathbb{Z}_{4}-$ graphs on 4 vertices.
Remark 4.5. It is proven in [5], Chapter 6, that any representation of $B_{4}$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible.
5. Representations whose friendship graph is a chain
Definition 5.1. The standard representation is the representation
$$
\tau_{n}:B_{n}\to G L_{n}(\mathbb{Z}[t^{\pm1}]
$$
defined by
$$
\tau_{n}(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&&&\\ &{0}&{t}&\\ &{1}&{0}&\\ &&&{I_{n-1-i}}\end{array}\right),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix.
Theorem 5.1. Let $\rho:B_{n}\to G L_{n}(\mathbb{C})$ be an irreducible representation, where $n\geq4$ . Suppose that $\rho(\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\rho$ is a chain.
Then $\rho$ is equivalent to a specialization $\tau_{n}(u)$ of the standard representation for some $u\in\mathbb{C}^{*}$ .
Before proving the theorem, we will need the following technical lemma:
Lemma 5.2. Let A be a friend and a neighbor of B, $B$ be a friend and a neighbor of $C$ and suppose that A is not a friend of $C$ :
|
<html><body>
<p data-bbox="123 110 487 138">4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case. </p>
<p data-bbox="124 139 486 181">3) If $n=4$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $\mathbb{Z}_{4}-$ graphs on 4 vertices. </p>
<p data-bbox="124 187 487 230">Remark 4.5. It is proven in [5], Chapter 6, that any representation of $B_{4}$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible. </p>
<p data-bbox="140 246 471 259">5. Representations whose friendship graph is a chain </p>
<p data-bbox="124 265 486 293">Definition 5.1. The standard representation is the representation </p>
<div class="equation" data-bbox="249 296 362 312">$$
\tau_{n}:B_{n}\to G L_{n}(\mathbb{Z}[t^{\pm1}]
$$</div>
<p data-bbox="125 314 180 329">defined by </p>
<div class="equation" data-bbox="219 369 391 428">$$
\tau_{n}(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&&&\\ &{0}&{t}&\\ &{1}&{0}&\\ &&&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="124 442 434 458">for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix. </p>
<p data-bbox="125 472 487 515">Theorem 5.1. Let $\rho:B_{n}\to G L_{n}(\mathbb{C})$ be an irreducible representation, where $n\geq4$ . Suppose that $\rho(\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\rho$ is a chain. </p>
<p data-bbox="126 516 485 543">Then $\rho$ is equivalent to a specialization $\tau_{n}(u)$ of the standard representation for some $u\in\mathbb{C}^{*}$ . </p>
<div class="image" data-bbox="124 567 404 607"><img data-bbox="124 567 404 607"/></div>
<p data-bbox="123 635 487 664">Before proving the theorem, we will need the following technical lemma: </p>
<p data-bbox="124 671 487 700">Lemma 5.2. Let A be a friend and a neighbor of B, $B$ be a friend and a neighbor of $C$ and suppose that A is not a friend of $C$ : </p>
</body></html>
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0003047v1
| 11 | 612 | 792 | 1,275 | 1,650 |
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{"category_id": 15, "poly": [797.0, 1237.0, 894.0, 1237.0, 894.0, 1278.0, 797.0, 1278.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [974.0, 1237.0, 1161.0, 1237.0, 1161.0, 1278.0, 974.0, 1278.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 11, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 486, 127], "score": 1.0, "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [278, 92, 329, 102], "spans": [{"bbox": [278, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 90, 136, 100], "lines": [{"bbox": [125, 92, 137, 103], "spans": [{"bbox": [125, 92, 137, 103], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 486, 127], "score": 1.0, "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 113, 486, 140]}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [125, 140, 487, 182]}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6, "bbox_fs": [124, 190, 486, 231]}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [141, 248, 470, 261]}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 268, 486, 297]}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [126, 314, 179, 331]}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [125, 445, 434, 459]}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [126, 474, 488, 516]}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [127, 516, 485, 544]}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5, "bbox_fs": [125, 637, 486, 665]}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 674, 487, 702]}]}
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[{"type": "text", "bbox": [123, 110, 487, 138], "content": "4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.", "index": 0}, {"type": "text", "bbox": [124, 139, 486, 181], "content": "3) If , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.", "index": 1}, {"type": "text", "bbox": [124, 187, 487, 230], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation of with either of the exeptional friendship graphs in 3) of the above theorem is reducible.", "index": 2}, {"type": "text", "bbox": [140, 246, 471, 259], "content": "5. Representations whose friendship graph is a chain", "index": 3}, {"type": "text", "bbox": [124, 265, 486, 293], "content": "Definition 5.1. The standard representation is the representa- tion", "index": 4}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "content": "", "index": 5}, {"type": "text", "bbox": [125, 314, 180, 329], "content": "defined by", "index": 6}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "content": "", "index": 7}, {"type": "text", "bbox": [124, 442, 434, 458], "content": "for , where is the identity matrix.", "index": 8}, {"type": "text", "bbox": [125, 472, 487, 515], "content": "Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.", "index": 9}, {"type": "text", "bbox": [126, 516, 485, 543], "content": "Then is equivalent to a specialization of the standard repre- sentation for some .", "index": 10}, {"type": "image", "bbox": [124, 567, 404, 607], "content": "", "index": 11}, {"type": "text", "bbox": [123, 635, 487, 664], "content": "Before proving the theorem, we will need the following technical lemma:", "index": 12}, {"type": "text", "bbox": [124, 671, 487, 700], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, be a friend and a neighbor of and suppose that A is not a friend of :", "index": 13}]
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[{"bbox": [126, 113, 486, 127], "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 269, 140], "content": "obtain the exceptional case.", "parent_index": 0, "line_index": 1}, {"bbox": [137, 140, 484, 155], "content": "3) If , then by theorem 3.8 the friendship graph is not totally", "parent_index": 1, "line_index": 0}, {"bbox": [125, 154, 487, 169], "content": "disconnected. Hence, we have only three possible graphs on 4", "parent_index": 1, "line_index": 1}, {"bbox": [126, 170, 168, 182], "content": "vertices.", "parent_index": 1, "line_index": 2}, {"bbox": [124, 190, 486, 206], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "parent_index": 2, "line_index": 0}, {"bbox": [125, 204, 486, 218], "content": "of with either of the exeptional friendship graphs in 3) of the above", "parent_index": 2, "line_index": 1}, {"bbox": [126, 219, 232, 231], "content": "theorem is reducible.", "parent_index": 2, "line_index": 2}, {"bbox": [141, 248, 470, 261], "content": "5. Representations whose friendship graph is a chain", "parent_index": 3, "line_index": 0}, {"bbox": [125, 268, 486, 282], "content": "Definition 5.1. The standard representation is the representa-", "parent_index": 4, "line_index": 0}, {"bbox": [125, 282, 150, 297], "content": "tion", "parent_index": 4, "line_index": 1}, {"bbox": [126, 314, 179, 331], "content": "defined by", "parent_index": 6, "line_index": 0}, {"bbox": [125, 445, 434, 459], "content": "for , where is the identity matrix.", "parent_index": 8, "line_index": 0}, {"bbox": [126, 474, 486, 489], "content": "Theorem 5.1. Let be an irreducible representation,", "parent_index": 9, "line_index": 0}, {"bbox": [126, 489, 488, 504], "content": "where . Suppose that , where , and", "parent_index": 9, "line_index": 1}, {"bbox": [126, 503, 365, 516], "content": "the associated friendship graph of is a chain.", "parent_index": 9, "line_index": 2}, {"bbox": [139, 516, 485, 532], "content": "Then is equivalent to a specialization of the standard repre-", "parent_index": 10, "line_index": 0}, {"bbox": [127, 532, 264, 544], "content": "sentation for some .", "parent_index": 10, "line_index": 1}, {"bbox": [138, 637, 486, 651], "content": "Before proving the theorem, we will need the following technical", "parent_index": 12, "line_index": 0}, {"bbox": [125, 651, 164, 665], "content": "lemma:", "parent_index": 12, "line_index": 1}, {"bbox": [125, 674, 487, 687], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, be a friend and", "parent_index": 13, "line_index": 0}, {"bbox": [126, 687, 423, 702], "content": "a neighbor of and suppose that A is not a friend of :", "parent_index": 13, "line_index": 1}]
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[{"bbox": [124, 567, 404, 607], "content": "", "parent_index": 11, "subtype": "body"}]
|
[{"bbox": [164, 143, 194, 151], "content": "n=4", "parent_index": 1, "subtype": "inline"}, {"bbox": [398, 156, 421, 167], "content": "\\mathbb{Z}_{4}-", "parent_index": 1, "subtype": "inline"}, {"bbox": [139, 206, 153, 217], "content": "B_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [249, 296, 362, 312], "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "parent_index": 5, "subtype": "interline"}, {"bbox": [219, 369, 391, 428], "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 7, "subtype": "interline"}, {"bbox": [144, 447, 235, 458], "content": "i=1,2,\\dots,n-1", "parent_index": 8, "subtype": "inline"}, {"bbox": [275, 446, 286, 457], "content": "I_{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [322, 447, 350, 457], "content": "k\\times k", "parent_index": 8, "subtype": "inline"}, {"bbox": [229, 475, 319, 488], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "parent_index": 9, "subtype": "inline"}, {"bbox": [159, 489, 191, 501], "content": "n\\geq4", "parent_index": 9, "subtype": "inline"}, {"bbox": [267, 489, 346, 502], "content": "\\rho(\\sigma_{1})=1+A_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [387, 490, 459, 503], "content": "r a n k(A_{1})=2", "parent_index": 9, "subtype": "inline"}, {"bbox": [300, 505, 307, 516], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [168, 519, 176, 530], "content": "\\rho", "parent_index": 10, "subtype": "inline"}, {"bbox": [342, 518, 370, 531], "content": "\\tau_{n}(u)", "parent_index": 10, "subtype": "inline"}, {"bbox": [225, 532, 260, 542], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 10, "subtype": "inline"}, {"bbox": [396, 676, 406, 684], "content": "B", "parent_index": 13, "subtype": "inline"}, {"bbox": [197, 690, 206, 699], "content": "C", "parent_index": 13, "subtype": "inline"}, {"bbox": [403, 690, 414, 699], "content": "C", "parent_index": 13, "subtype": "inline"}]
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[]
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Let $a~\ne~0$ be such that s $\L)a n\{a\}\;=\;I m(A)\cap I m(B)$ , and let $b\,=$ $(1+B)a$ . Then:
1) $s p a n\{b\}=I m(C)\cap I m(B)$ .
$\mathcal{Q}_{g}$ ) $(1+B)b\in s p a n\{a\}$ and $(1+B)b\neq0$ .
3) The vectors $a$ and $b$ are linearly independent.
Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\ne0$ .
1) $b=(1+B)a\in I m(B)$ , because $a\in I m(B)$ .
$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then
$$
(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=
$$
$$
=(C+C^{2}+C B C)a_{1}\in I m(C);
$$
that is, $b\;\in\;I m(C)\cap I m(B)$ , and because $I m(C)\cap I m(B)$ is onedimensional and $b\neq0$ ,
$$
s p a n\{b\}=I m(C)\cap I m(B).
$$
2) Clearly, $(1+B)b\in I m(B)$ .
Note, that $A b=0$ , as $b\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then
$$
(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\in I m(A).
$$
3) $a\in I m(A)$ , $b\in I m(C)$ by part 1), and $I m(A)\cap I m(C)=\{0\}$ by the hypothesis of the lemma.
Proof of Theorem 5.1 We include the redundant generator $\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\cap I m(A_{i+1})$ , which is $\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then
$$
I m(A_{0})=I m(A_{1})=\cdots=I m(A_{n-1})
$$
would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence, $I m(A_{i})\cap I m(A_{i+1})$ is one-dimensional.
Let $a_{0}$ be a basis vector for $I m(A_{0})\cap I m(A_{1})$ . Let
$$
a_{1}=(1+A_{1})a_{0},\;\;a_{2}=(1+A_{2})a_{1},\;\;.\;.\;.\;,\;\;a_{n-1}=(1+A_{n-1})a_{n-2}.
$$
|
<html><body>
<p data-bbox="124 166 487 195">Let $a~\ne~0$ be such that s $\L)a n\{a\}\;=\;I m(A)\cap I m(B)$ , and let $b\,=$ $(1+B)a$ . Then: </p>
<p data-bbox="136 195 385 237">1) $s p a n\{b\}=I m(C)\cap I m(B)$ .
$\mathcal{Q}_{g}$ ) $(1+B)b\in s p a n\{a\}$ and $(1+B)b\neq0$ .
3) The vectors $a$ and $b$ are linearly independent. </p>
<p data-bbox="126 245 486 273">Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\ne0$ . </p>
<p data-bbox="136 273 378 288">1) $b=(1+B)a\in I m(B)$ , because $a\in I m(B)$ . </p>
<p data-bbox="124 288 485 315">$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then </p>
<div class="equation" data-bbox="126 324 488 339">$$
(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=
$$</div>
<div class="equation" data-bbox="221 349 387 363">$$
=(C+C^{2}+C B C)a_{1}\in I m(C);
$$</div>
<p data-bbox="124 366 485 394">that is, $b\;\in\;I m(C)\cap I m(B)$ , and because $I m(C)\cap I m(B)$ is onedimensional and $b\neq0$ , </p>
<div class="equation" data-bbox="233 403 377 417">$$
s p a n\{b\}=I m(C)\cap I m(B).
$$</div>
<p data-bbox="136 423 292 437">2) Clearly, $(1+B)b\in I m(B)$ . </p>
<p data-bbox="123 438 487 466">Note, that $A b=0$ , as $b\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then </p>
<div class="equation" data-bbox="126 473 495 489">$$
(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\in I m(A).
$$</div>
<p data-bbox="124 494 487 523">3) $a\in I m(A)$ , $b\in I m(C)$ by part 1), and $I m(A)\cap I m(C)=\{0\}$ by the hypothesis of the lemma. </p>
<p data-bbox="124 551 487 607">Proof of Theorem 5.1 We include the redundant generator $\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\cap I m(A_{i+1})$ , which is $\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then </p>
<div class="equation" data-bbox="210 617 401 630">$$
I m(A_{0})=I m(A_{1})=\cdots=I m(A_{n-1})
$$</div>
<p data-bbox="124 636 487 664">would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence, $I m(A_{i})\cap I m(A_{i+1})$ is one-dimensional. </p>
<p data-bbox="135 665 399 679">Let $a_{0}$ be a basis vector for $I m(A_{0})\cap I m(A_{1})$ . Let </p>
<div class="equation" data-bbox="141 688 470 702">$$
a_{1}=(1+A_{1})a_{0},\;\;a_{2}=(1+A_{2})a_{1},\;\;.\;.\;.\;,\;\;a_{n-1}=(1+A_{n-1})a_{n-2}.
$$</div>
</body></html>
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0003047v1
| 12 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Let $a~\\ne~0$ be such that s $\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)$ , and let $b\\,=$ $(1+B)a$ . Then: ", "page_idx": 12}, {"type": "text", "text": "1) $s p a n\\{b\\}=I m(C)\\cap I m(B)$ . \n$\\mathcal{Q}_{g}$ ) $(1+B)b\\in s p a n\\{a\\}$ and $(1+B)b\\neq0$ . \n3) The vectors $a$ and $b$ are linearly independent. ", "page_idx": 12}, {"type": "text", "text": "Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\\ne0$ . ", "page_idx": 12}, {"type": "text", "text": "1) $b=(1+B)a\\in I m(B)$ , because $a\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=\n$$", "text_format": "latex", "page_idx": 12}, {"type": "equation", "text": "$$\n=(C+C^{2}+C B C)a_{1}\\in I m(C);\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "that is, $b\\;\\in\\;I m(C)\\cap I m(B)$ , and because $I m(C)\\cap I m(B)$ is onedimensional and $b\\neq0$ , ", "page_idx": 12}, {"type": "equation", "text": "$$\ns p a n\\{b\\}=I m(C)\\cap I m(B).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "2) Clearly, $(1+B)b\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "Note, that $A b=0$ , as $b\\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "3) $a\\in I m(A)$ , $b\\in I m(C)$ by part 1), and $I m(A)\\cap I m(C)=\\{0\\}$ by the hypothesis of the lemma. ", "page_idx": 12}, {"type": "text", "text": "Proof of Theorem 5.1 We include the redundant generator $\\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\\cap I m(A_{i+1})$ , which is $\\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then ", "page_idx": 12}, {"type": "equation", "text": "$$\nI m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence, $I m(A_{i})\\cap I m(A_{i+1})$ is one-dimensional. ", "page_idx": 12}, {"type": "text", "text": "Let $a_{0}$ be a basis vector for $I m(A_{0})\\cap I m(A_{1})$ . Let ", "page_idx": 12}, {"type": "equation", "text": "$$\na_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.\n$$", "text_format": "latex", "page_idx": 12}]
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First of all, notice that the vector ", "type": "text"}, {"bbox": [369, 248, 375, 258], "score": 0.8, "content": "b", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [376, 247, 486, 261], "score": 1.0, "content": " is non-zero, because", "type": "text"}], "index": 5}, {"bbox": [126, 262, 277, 274], "spans": [{"bbox": [126, 263, 156, 273], "score": 0.91, "content": "1+B", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 262, 245, 273], "score": 1.0, "content": " is invertible and ", "type": "text"}, {"bbox": [246, 263, 274, 274], "score": 0.89, "content": "a\\ne0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [274, 262, 277, 273], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [136, 273, 378, 288], "lines": [{"bbox": [138, 275, 377, 290], "spans": [{"bbox": [138, 275, 152, 290], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [152, 276, 268, 289], "score": 0.92, "content": "b=(1+B)a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [268, 275, 318, 290], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [318, 276, 374, 289], "score": 0.93, "content": "a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [374, 275, 377, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 288, 485, 315], "lines": [{"bbox": [138, 288, 486, 303], "spans": [{"bbox": [138, 291, 147, 300], "score": 0.86, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [147, 288, 174, 303], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 291, 183, 300], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 288, 308, 303], "score": 1.0, "content": " are not friends, that is ", "type": "text"}, {"bbox": [308, 291, 349, 300], "score": 0.9, "content": "C A=0", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [349, 288, 370, 303], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 290, 410, 300], "score": 0.88, "content": "C a=0", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [410, 288, 437, 303], "score": 1.0, "content": ". 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Consider ", "type": "text"}, {"bbox": [311, 568, 411, 581], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [411, 567, 462, 581], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [462, 570, 469, 578], "score": 0.45, "content": "\\boldsymbol{0}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [469, 567, 475, 581], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [476, 570, 482, 578], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [482, 567, 485, 581], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [126, 582, 486, 595], "spans": [{"bbox": [126, 582, 141, 595], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [141, 583, 156, 593], "score": 0.88, "content": "2-", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [157, 582, 486, 595], "score": 1.0, "content": "dimensional. It is nonzero, because of the hypothesis that the", "type": "text"}], "index": 23}, {"bbox": [126, 595, 435, 608], "spans": [{"bbox": [126, 595, 435, 608], "score": 1.0, "content": "friendship graph is a chain. 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First of all, notice that the vector ", "type": "text"}, {"bbox": [369, 248, 375, 258], "score": 0.8, "content": "b", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [376, 247, 486, 261], "score": 1.0, "content": " is non-zero, because", "type": "text"}], "index": 5}, {"bbox": [126, 262, 277, 274], "spans": [{"bbox": [126, 263, 156, 273], "score": 0.91, "content": "1+B", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 262, 245, 273], "score": 1.0, "content": " is invertible and ", "type": "text"}, {"bbox": [246, 263, 274, 274], "score": 0.89, "content": "a\\ne0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [274, 262, 277, 273], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [126, 247, 486, 274]}, {"type": "text", "bbox": [136, 273, 378, 288], "lines": [{"bbox": [138, 275, 377, 290], "spans": [{"bbox": [138, 275, 152, 290], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [152, 276, 268, 289], "score": 0.92, "content": "b=(1+B)a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [268, 275, 318, 290], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [318, 276, 374, 289], "score": 0.93, "content": "a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [374, 275, 377, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [138, 275, 377, 290]}, {"type": "text", "bbox": [124, 288, 485, 315], "lines": [{"bbox": [138, 288, 486, 303], "spans": [{"bbox": [138, 291, 147, 300], "score": 0.86, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [147, 288, 174, 303], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 291, 183, 300], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 288, 308, 303], "score": 1.0, "content": " are not friends, that is ", "type": "text"}, {"bbox": [308, 291, 349, 300], "score": 0.9, "content": "C A=0", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [349, 288, 370, 303], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 290, 410, 300], "score": 0.88, "content": "C a=0", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [410, 288, 437, 303], "score": 1.0, "content": ". 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Consider ", "type": "text"}, {"bbox": [311, 568, 411, 581], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [411, 567, 462, 581], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [462, 570, 469, 578], "score": 0.45, "content": "\\boldsymbol{0}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [469, 567, 475, 581], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [476, 570, 482, 578], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [482, 567, 485, 581], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [126, 582, 486, 595], "spans": [{"bbox": [126, 582, 141, 595], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [141, 583, 156, 593], "score": 0.88, "content": "2-", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [157, 582, 486, 595], "score": 1.0, "content": "dimensional. It is nonzero, because of the hypothesis that the", "type": "text"}], "index": 23}, {"bbox": [126, 595, 435, 608], "spans": [{"bbox": [126, 595, 435, 608], "score": 1.0, "content": "friendship graph is a chain. It is not 2-dimensional, for then", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [125, 550, 487, 608]}, {"type": "interline_equation", "bbox": [210, 617, 401, 630], "lines": [{"bbox": [210, 617, 401, 630], "spans": [{"bbox": [210, 617, 401, 630], "score": 0.89, "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [124, 636, 487, 664], "lines": [{"bbox": [126, 639, 484, 651], "spans": [{"bbox": [126, 639, 187, 651], "score": 1.0, "content": "would be a ", "type": "text"}, {"bbox": [188, 641, 203, 650], "score": 0.89, "content": "2-", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [203, 639, 484, 651], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irre-", "type": "text"}], "index": 26}, {"bbox": [127, 653, 444, 666], "spans": [{"bbox": [127, 653, 191, 666], "score": 1.0, "content": "ducibility of ", "type": "text"}, {"bbox": [191, 657, 198, 665], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [198, 653, 242, 666], "score": 1.0, "content": ". 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Let", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [137, 666, 397, 680]}, {"type": "interline_equation", "bbox": [141, 688, 470, 702], "lines": [{"bbox": [141, 688, 470, 702], "spans": [{"bbox": [141, 688, 470, 702], "score": 0.89, "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "type": "interline_equation"}], "index": 29}], "index": 29}]}
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[{"type": "text", "bbox": [124, 166, 487, 195], "content": "Let be such that s , and let . Then:", "index": 0}, {"type": "list", "bbox": [136, 195, 385, 237], "content": "", "index": 1}, {"type": "text", "bbox": [126, 245, 486, 273], "content": "Proof. First of all, notice that the vector is non-zero, because is invertible and .", "index": 2}, {"type": "text", "bbox": [136, 273, 378, 288], "content": "1) , because .", "index": 3}, {"type": "text", "bbox": [124, 288, 485, 315], "content": "and are not friends, that is , so . Let . Then", "index": 4}, {"type": "interline_equation", "bbox": [126, 324, 488, 339], "content": "", "index": 5}, {"type": "interline_equation", "bbox": [221, 349, 387, 363], "content": "", "index": 6}, {"type": "text", "bbox": [124, 366, 485, 394], "content": "that is, , and because is one- dimensional and ,", "index": 7}, {"type": "interline_equation", "bbox": [233, 403, 377, 417], "content": "", "index": 8}, {"type": "text", "bbox": [136, 423, 292, 437], "content": "2) Clearly, .", "index": 9}, {"type": "text", "bbox": [123, 438, 487, 466], "content": "Note, that , as by the above, and . Let . Then", "index": 10}, {"type": "interline_equation", "bbox": [126, 473, 495, 489], "content": "", "index": 11}, {"type": "text", "bbox": [124, 494, 487, 523], "content": "3) , by part 1), and by the hypothesis of the lemma.", "index": 12}, {"type": "text", "bbox": [124, 551, 487, 607], "content": "Proof of Theorem 5.1 We include the redundant generator , and indices are modulo . Consider , which is , , or dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then", "index": 13}, {"type": "interline_equation", "bbox": [210, 617, 401, 630], "content": "", "index": 14}, {"type": "text", "bbox": [124, 636, 487, 664], "content": "would be a dimensional invariant subspace, contradicting the irre- ducibility of . Hence, is one-dimensional.", "index": 15}, {"type": "text", "bbox": [135, 665, 399, 679], "content": "Let be a basis vector for . Let", "index": 16}, {"type": "interline_equation", "bbox": [141, 688, 470, 702], "content": "", "index": 17}]
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[{"bbox": [137, 168, 486, 183], "content": "Let be such that s , and let", "parent_index": 0, "line_index": 0}, {"bbox": [126, 182, 212, 197], "content": ". Then:", "parent_index": 0, "line_index": 1}, {"bbox": [139, 198, 297, 211], "content": "1) .", "parent_index": 1, "line_index": 0}, {"bbox": [138, 211, 349, 225], "content": ") and .", "parent_index": 1, "line_index": 1}, {"bbox": [139, 225, 383, 237], "content": "3) The vectors and are linearly independent.", "parent_index": 1, "line_index": 2}, {"bbox": [137, 247, 486, 261], "content": "Proof. First of all, notice that the vector is non-zero, because", "parent_index": 2, "line_index": 0}, {"bbox": [126, 262, 277, 274], "content": "is invertible and .", "parent_index": 2, "line_index": 1}, {"bbox": [138, 275, 377, 290], "content": "1) , because .", "parent_index": 3, "line_index": 0}, {"bbox": [138, 288, 486, 303], "content": "and are not friends, that is , so . Let .", "parent_index": 4, "line_index": 0}, {"bbox": [124, 302, 155, 317], "content": "Then", "parent_index": 4, "line_index": 1}, {"bbox": [126, 367, 485, 383], "content": "that is, , and because is one-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 382, 244, 396], "content": "dimensional and ,", "parent_index": 7, "line_index": 1}, {"bbox": [137, 424, 292, 439], "content": "2) Clearly, .", "parent_index": 9, "line_index": 0}, {"bbox": [136, 437, 487, 454], "content": "Note, that , as by the above, and . Let", "parent_index": 10, "line_index": 0}, {"bbox": [126, 452, 201, 467], "content": ". Then", "parent_index": 10, "line_index": 1}, {"bbox": [137, 496, 484, 512], "content": "3) , by part 1), and by", "parent_index": 12, "line_index": 0}, {"bbox": [126, 511, 275, 524], "content": "the hypothesis of the lemma.", "parent_index": 12, "line_index": 1}, {"bbox": [135, 550, 487, 570], "content": "Proof of Theorem 5.1 We include the redundant generator ,", "parent_index": 13, "line_index": 0}, {"bbox": [125, 567, 485, 581], "content": "and indices are modulo . Consider , which is , ,", "parent_index": 13, "line_index": 1}, {"bbox": [126, 582, 486, 595], "content": "or dimensional. It is nonzero, because of the hypothesis that the", "parent_index": 13, "line_index": 2}, {"bbox": [126, 595, 435, 608], "content": "friendship graph is a chain. It is not 2-dimensional, for then", "parent_index": 13, "line_index": 3}, {"bbox": [126, 639, 484, 651], "content": "would be a dimensional invariant subspace, contradicting the irre-", "parent_index": 15, "line_index": 0}, {"bbox": [127, 653, 444, 666], "content": "ducibility of . Hence, is one-dimensional.", "parent_index": 15, "line_index": 1}, {"bbox": [137, 666, 397, 680], "content": "Let be a basis vector for . Let", "parent_index": 16, "line_index": 0}]
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[]
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[{"bbox": [159, 170, 192, 182], "content": "a~\\ne~0", "parent_index": 0, "subtype": "inline"}, {"bbox": [275, 169, 415, 183], "content": "\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)", "parent_index": 0, "subtype": "inline"}, {"bbox": [465, 169, 486, 182], "content": "b\\,=", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 184, 171, 197], "content": "(1+B)a", "parent_index": 0, "subtype": "inline"}, {"bbox": [153, 198, 294, 210], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B)", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 212, 145, 223], "content": "\\mathcal{Q}_{g}", "parent_index": 1, "subtype": "inline"}, {"bbox": [153, 212, 254, 225], "content": "(1+B)b\\in s p a n\\{a\\}", "parent_index": 1, "subtype": "inline"}, {"bbox": [281, 211, 347, 225], "content": "(1+B)b\\neq0", "parent_index": 1, "subtype": "inline"}, {"bbox": [216, 228, 223, 236], "content": "a", "parent_index": 1, "subtype": "inline"}, {"bbox": [249, 226, 255, 235], "content": "b", "parent_index": 1, "subtype": "inline"}, {"bbox": [369, 248, 375, 258], "content": "b", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 263, 156, 273], "content": "1+B", "parent_index": 2, "subtype": "inline"}, {"bbox": [246, 263, 274, 274], "content": "a\\ne0", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 276, 268, 289], "content": "b=(1+B)a\\in I m(B)", "parent_index": 3, "subtype": "inline"}, {"bbox": [318, 276, 374, 289], "content": "a\\in I m(B)", "parent_index": 3, "subtype": "inline"}, {"bbox": [138, 291, 147, 300], "content": "A", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 291, 183, 300], "content": "C", "parent_index": 4, "subtype": "inline"}, {"bbox": [308, 291, 349, 300], "content": "C A=0", "parent_index": 4, "subtype": "inline"}, {"bbox": [371, 290, 410, 300], "content": "C a=0", "parent_index": 4, "subtype": "inline"}, {"bbox": [438, 290, 482, 302], "content": "a=B a_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 324, 488, 339], "content": "(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=", "parent_index": 5, "subtype": "interline"}, {"bbox": [221, 349, 387, 363], "content": "=(C+C^{2}+C B C)a_{1}\\in I m(C);", "parent_index": 6, "subtype": "interline"}, {"bbox": [170, 370, 281, 382], "content": "b\\;\\in\\;I m(C)\\cap I m(B)", "parent_index": 7, "subtype": "inline"}, {"bbox": [358, 367, 444, 382], "content": "I m(C)\\cap I m(B)", "parent_index": 7, "subtype": "inline"}, {"bbox": [213, 384, 240, 395], "content": "b\\neq0", "parent_index": 7, "subtype": "inline"}, {"bbox": [233, 403, 377, 417], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B).", "parent_index": 8, "subtype": "interline"}, {"bbox": [195, 426, 289, 439], "content": "(1+B)b\\in I m(B)", "parent_index": 9, "subtype": "inline"}, {"bbox": [196, 441, 235, 450], "content": "A b=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 439, 315, 453], "content": "b\\in I m(C)", "parent_index": 10, "subtype": "inline"}, {"bbox": [417, 439, 460, 450], "content": "A C=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 453, 165, 464], "content": "b=B a^{'}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 473, 495, 489], "content": "(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).", "parent_index": 11, "subtype": "interline"}, {"bbox": [152, 498, 207, 510], "content": "a\\in I m(A)", "parent_index": 12, "subtype": "inline"}, {"bbox": [214, 498, 268, 511], "content": "b\\in I m(C)", "parent_index": 12, "subtype": "inline"}, {"bbox": [353, 498, 469, 511], "content": "I m(A)\\cap I m(C)=\\{0\\}", "parent_index": 12, "subtype": "inline"}, {"bbox": [470, 558, 482, 565], "content": "\\sigma_{0}", "parent_index": 13, "subtype": "inline"}, {"bbox": [248, 572, 255, 578], "content": "n", "parent_index": 13, "subtype": "inline"}, {"bbox": [311, 568, 411, 581], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 13, "subtype": "inline"}, {"bbox": [462, 570, 469, 578], "content": "\\boldsymbol{0}", "parent_index": 13, "subtype": "inline"}, {"bbox": [476, 570, 482, 578], "content": "1", "parent_index": 13, "subtype": "inline"}, {"bbox": [141, 583, 156, 593], "content": "2-", "parent_index": 13, "subtype": "inline"}, {"bbox": [210, 617, 401, 630], "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "parent_index": 14, "subtype": "interline"}, {"bbox": [188, 641, 203, 650], "content": "2-", "parent_index": 15, "subtype": "inline"}, {"bbox": [191, 657, 198, 665], "content": "\\rho", "parent_index": 15, "subtype": "inline"}, {"bbox": [242, 653, 342, 666], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 15, "subtype": "inline"}, {"bbox": [159, 671, 170, 679], "content": "a_{0}", "parent_index": 16, "subtype": "inline"}, {"bbox": [281, 667, 373, 680], "content": "I m(A_{0})\\cap I m(A_{1})", "parent_index": 16, "subtype": "inline"}, {"bbox": [141, 688, 470, 702], "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "parent_index": 17, "subtype": "interline"}]
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[]
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By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ , for $0\leq\,i\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\{a_{i},a_{i+1}\}$ is a basis for $I m(A_{i})$ .
Since
$$
s p a n\{a_{0},\ldots a_{n-1}\}=I m(A_{1})+\cdots+I m(A_{n-1})
$$
is invariant under $B_{n}$ and $\rho$ is an $n-$ dimensional irreducible representation, $\left\{a_{0},\ldots.a_{n-1}\right\}$ is a basis for $\mathbb{C}^{n}$ .
We now wish to determine the action of $\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$ on this basis.
Consider $a_{i}\in I m(A_{i})\cap I m(A_{i+1})$ . If $j\neq i,\ \ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and
$$
\rho(\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.
$$
By our construction
$$
\rho(\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}
$$
for $0\leq i\leq n-2$ .
By lemma 5.2, part 2),
$$
\rho(\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},
$$
for $1\leq i\leq n-1$ , where $u_{i}\in\mathbb{C}^{*}$ .
By the above calculations the matrices of $\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$ with respect to the basis $a_{0},\;\;a_{1},\ldots,a_{n-1}$ are
$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u_{i}}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$
for $i\;=\;1,2,\ldots,n\,-\,1$ , where $I_{k}$ is the $k\,\times\,k$ identity matrix, and $u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{*}$ . Since $\sigma_{1},\ldots,\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation.
Now let us consider when the standard representation is irreducible.
Lemma 5.3. If $u=1$ then $\tau_{n}(u)$ is reducible.
Proof. If $u=1$ then the vector $v=(1,1,1,\ldots,1)^{T}$ is a fixed vector.
Lemma 5.4. If $u\ne1$ then $\tau_{n}(u)$ is irreducible.
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<html><body>
<p data-bbox="124 110 487 153">By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ , for $0\leq\,i\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\{a_{i},a_{i+1}\}$ is a basis for $I m(A_{i})$ . </p>
<p data-bbox="137 153 167 165">Since </p>
<div class="equation" data-bbox="186 174 425 187">$$
s p a n\{a_{0},\ldots a_{n-1}\}=I m(A_{1})+\cdots+I m(A_{n-1})
$$</div>
<p data-bbox="123 191 485 218">is invariant under $B_{n}$ and $\rho$ is an $n-$ dimensional irreducible representation, $\left\{a_{0},\ldots.a_{n-1}\right\}$ is a basis for $\mathbb{C}^{n}$ . </p>
<p data-bbox="124 219 486 245">We now wish to determine the action of $\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$ on this basis. </p>
<p data-bbox="124 246 487 288">Consider $a_{i}\in I m(A_{i})\cap I m(A_{i+1})$ . If $j\neq i,\ \ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and </p>
<div class="equation" data-bbox="239 295 371 309">$$
\rho(\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.
$$</div>
<p data-bbox="137 312 242 325">By our construction </p>
<div class="equation" data-bbox="225 333 385 346">$$
\rho(\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}
$$</div>
<p data-bbox="124 349 217 363">for $0\leq i\leq n-2$ . </p>
<p data-bbox="137 364 255 377">By lemma 5.2, part 2), </p>
<div class="equation" data-bbox="229 384 380 398">$$
\rho(\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},
$$</div>
<p data-bbox="124 401 297 415">for $1\leq i\leq n-1$ , where $u_{i}\in\mathbb{C}^{*}$ . </p>
<p data-bbox="124 416 486 444">By the above calculations the matrices of $\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$ with respect to the basis $a_{0},\;\;a_{1},\ldots,a_{n-1}$ are </p>
<div class="equation" data-bbox="219 483 390 540">$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u_{i}}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="123 553 487 596">for $i\;=\;1,2,\ldots,n\,-\,1$ , where $I_{k}$ is the $k\,\times\,k$ identity matrix, and $u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{*}$ . Since $\sigma_{1},\ldots,\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation. </p>
<p data-bbox="135 609 485 624">Now let us consider when the standard representation is irreducible. </p>
<p data-bbox="124 630 362 645">Lemma 5.3. If $u=1$ then $\tau_{n}(u)$ is reducible. </p>
<p data-bbox="136 650 486 666">Proof. If $u=1$ then the vector $v=(1,1,1,\ldots,1)^{T}$ is a fixed vector. </p>
<p data-bbox="124 685 372 701">Lemma 5.4. If $u\ne1$ then $\tau_{n}(u)$ is irreducible. </p>
</body></html>
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0003047v1
| 13 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ , for $0\\leq\\,i\\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\\{a_{i},a_{i+1}\\}$ is a basis for $I m(A_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "Since ", "page_idx": 13}, {"type": "equation", "text": "$$\ns p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "is invariant under $B_{n}$ and $\\rho$ is an $n-$ dimensional irreducible representation, $\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}$ is a basis for $\\mathbb{C}^{n}$ . ", "page_idx": 13}, {"type": "text", "text": "We now wish to determine the action of $\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})$ on this basis. ", "page_idx": 13}, {"type": "text", "text": "Consider $a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})$ . If $j\\neq i,\\ \\ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "By our construction ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $0\\leq i\\leq n-2$ . ", "page_idx": 13}, {"type": "text", "text": "By lemma 5.2, part 2), ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $1\\leq i\\leq n-1$ , where $u_{i}\\in\\mathbb{C}^{*}$ . ", "page_idx": 13}, {"type": "text", "text": "By the above calculations the matrices of $\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})$ with respect to the basis $a_{0},\\;\\;a_{1},\\ldots,a_{n-1}$ are ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $i\\;=\\;1,2,\\ldots,n\\,-\\,1$ , where $I_{k}$ is the $k\\,\\times\\,k$ identity matrix, and $u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}$ . Since $\\sigma_{1},\\ldots,\\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation. ", "page_idx": 13}, {"type": "text", "text": "Now let us consider when the standard representation is irreducible. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.3. If $u=1$ then $\\tau_{n}(u)$ is reducible. ", "page_idx": 13}, {"type": "text", "text": "Proof. If $u=1$ then the vector $v=(1,1,1,\\ldots,1)^{T}$ is a fixed vector. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.4. If $u\\ne1$ then $\\tau_{n}(u)$ is irreducible. ", "page_idx": 13}]
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If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. 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If ", "type": "text"}, {"bbox": [331, 250, 394, 261], "score": 0.81, "content": "j\\neq i,\\ \\ i+1", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [394, 248, 428, 263], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [428, 250, 441, 262], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [442, 248, 486, 263], "score": 1.0, "content": " is not a", "type": "text"}], "index": 9}, {"bbox": [126, 262, 487, 276], "spans": [{"bbox": [126, 262, 223, 276], "score": 1.0, "content": "neighbor of one of ", "type": "text"}, {"bbox": [224, 264, 236, 275], "score": 0.84, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [237, 262, 247, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 264, 270, 275], "score": 0.89, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [271, 262, 308, 276], "score": 1.0, "content": " (since ", "type": 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", and", "type": "text"}], "index": 11}], "index": 10, "bbox_fs": [126, 248, 487, 290]}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "lines": [{"bbox": [239, 295, 371, 309], "spans": [{"bbox": [239, 295, 371, 309], "score": 0.9, "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [137, 312, 242, 325], "lines": [{"bbox": [138, 314, 241, 326], "spans": [{"bbox": [138, 314, 241, 326], "score": 1.0, "content": "By our construction", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [138, 314, 241, 326]}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "lines": [{"bbox": [225, 333, 385, 346], "spans": [{"bbox": [225, 333, 385, 346], "score": 0.91, "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 349, 217, 363], "lines": [{"bbox": [124, 350, 217, 365], 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"spans": [{"bbox": [125, 402, 144, 417], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 405, 213, 415], "score": 0.92, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [213, 402, 254, 417], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [254, 405, 292, 415], "score": 0.93, "content": "u_{i}\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [293, 402, 296, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [125, 402, 296, 417]}, {"type": "text", "bbox": [124, 416, 486, 444], "lines": [{"bbox": [137, 416, 486, 432], "spans": [{"bbox": [137, 416, 365, 432], "score": 1.0, "content": "By the above calculations the matrices of ", "type": "text"}, {"bbox": [365, 417, 457, 430], "score": 0.92, "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [457, 416, 486, 432], "score": 1.0, "content": " with", "type": "text"}], "index": 19}, {"bbox": [125, 431, 333, 446], "spans": [{"bbox": [125, 431, 228, 446], "score": 1.0, "content": "respect to the basis ", "type": "text"}, {"bbox": [229, 435, 311, 444], "score": 0.51, "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "type": "inline_equation", "height": 9, "width": 82}, {"bbox": [311, 431, 333, 446], "score": 1.0, "content": " are", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [125, 416, 486, 446]}, {"type": "interline_equation", "bbox": [219, 483, 390, 540], "lines": [{"bbox": [219, 483, 390, 540], "spans": [{"bbox": [219, 483, 390, 540], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [123, 553, 487, 596], "lines": [{"bbox": [124, 555, 486, 570], "spans": [{"bbox": [124, 555, 145, 570], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [145, 556, 245, 569], "score": 0.9, "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [246, 555, 289, 570], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [289, 556, 300, 568], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [300, 555, 340, 570], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [340, 557, 372, 567], "score": 0.9, "content": "k\\,\\times\\,k", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [372, 555, 486, 570], "score": 1.0, "content": " identity matrix, and", "type": "text"}], "index": 22}, {"bbox": [126, 569, 487, 585], "spans": [{"bbox": [126, 570, 218, 583], "score": 0.9, "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [218, 569, 256, 585], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [256, 572, 318, 583], "score": 0.87, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [318, 569, 410, 585], "score": 1.0, "content": " are conjugate in ", "type": "text"}, {"bbox": [411, 572, 425, 582], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [426, 569, 454, 585], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [454, 575, 464, 582], "score": 0.9, "content": "u_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [465, 569, 487, 585], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [126, 584, 388, 598], "spans": [{"bbox": [126, 584, 388, 598], "score": 1.0, "content": "all equal, and we have the standard representation.", "type": "text"}], "index": 24}], "index": 23, "bbox_fs": [124, 555, 487, 598]}, {"type": "text", "bbox": [135, 609, 485, 624], "lines": [{"bbox": [137, 612, 485, 626], "spans": [{"bbox": [137, 612, 485, 626], "score": 1.0, "content": "Now let us consider when the standard representation is irreducible.", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [137, 612, 485, 626]}, {"type": "text", "bbox": [124, 630, 362, 645], "lines": [{"bbox": [125, 632, 362, 646], "spans": [{"bbox": [125, 632, 212, 646], "score": 1.0, "content": "Lemma 5.3. If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26, "bbox_fs": [125, 632, 362, 646]}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [136, 652, 485, 668]}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. If ", "type": "text"}, {"bbox": [212, 687, 242, 701], "score": 0.9, "content": "u\\ne1", "type": "inline_equation", "height": 14, "width": 30}, {"bbox": [242, 688, 270, 701], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [270, 687, 298, 702], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [299, 688, 370, 701], "score": 1.0, "content": " is irreducible.", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [125, 687, 370, 702]}]}
|
[{"type": "text", "bbox": [124, 110, 487, 153], "content": "By induction and lemma 5.2, part 1), is a basis vector for , for . By lemma 5.2, part 3), and are linearly independent. Thus is a basis for .", "index": 0}, {"type": "text", "bbox": [137, 153, 167, 165], "content": "Since", "index": 1}, {"type": "interline_equation", "bbox": [186, 174, 425, 187], "content": "", "index": 2}, {"type": "text", "bbox": [123, 191, 485, 218], "content": "is invariant under and is an dimensional irreducible represen- tation, is a basis for .", "index": 3}, {"type": "text", "bbox": [124, 219, 486, 245], "content": "We now wish to determine the action of on this basis.", "index": 4}, {"type": "text", "bbox": [124, 246, 487, 288], "content": "Consider . If , then is not a neighbor of one of , (since ), say , and then , so , and", "index": 5}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "content": "", "index": 6}, {"type": "text", "bbox": [137, 312, 242, 325], "content": "By our construction", "index": 7}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "content": "", "index": 8}, {"type": "text", "bbox": [124, 349, 217, 363], "content": "for .", "index": 9}, {"type": "text", "bbox": [137, 364, 255, 377], "content": "By lemma 5.2, part 2),", "index": 10}, {"type": "interline_equation", "bbox": [229, 384, 380, 398], "content": "", "index": 11}, {"type": "text", "bbox": [124, 401, 297, 415], "content": "for , where .", "index": 12}, {"type": "text", "bbox": [124, 416, 486, 444], "content": "By the above calculations the matrices of with respect to the basis are", "index": 13}, {"type": "interline_equation", "bbox": [219, 483, 390, 540], "content": "", "index": 14}, {"type": "text", "bbox": [123, 553, 487, 596], "content": "for , where is the identity matrix, and . Since are conjugate in , the are all equal, and we have the standard representation.", "index": 15}, {"type": "text", "bbox": [135, 609, 485, 624], "content": "Now let us consider when the standard representation is irreducible.", "index": 16}, {"type": "text", "bbox": [124, 630, 362, 645], "content": "Lemma 5.3. If then is reducible.", "index": 17}, {"type": "text", "bbox": [136, 650, 486, 666], "content": "Proof. If then the vector is a fixed vector.", "index": 18}, {"type": "text", "bbox": [124, 685, 372, 701], "content": "Lemma 5.4. If then is irreducible.", "index": 19}]
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[{"bbox": [125, 113, 486, 127], "content": "By induction and lemma 5.2, part 1), is a basis vector for", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 141], "content": ", for . By lemma 5.2, part 3), and are", "parent_index": 0, "line_index": 1}, {"bbox": [125, 141, 429, 154], "content": "linearly independent. Thus is a basis for .", "parent_index": 0, "line_index": 2}, {"bbox": [137, 154, 167, 168], "content": "Since", "parent_index": 1, "line_index": 0}, {"bbox": [124, 193, 484, 206], "content": "is invariant under and is an dimensional irreducible represen-", "parent_index": 3, "line_index": 0}, {"bbox": [125, 206, 320, 220], "content": "tation, is a basis for .", "parent_index": 3, "line_index": 1}, {"bbox": [137, 219, 484, 235], "content": "We now wish to determine the action of", "parent_index": 4, "line_index": 0}, {"bbox": [125, 234, 194, 247], "content": "on this basis.", "parent_index": 4, "line_index": 1}, {"bbox": [138, 248, 486, 263], "content": "Consider . If , then is not a", "parent_index": 5, "line_index": 0}, {"bbox": [126, 262, 487, 276], "content": "neighbor of one of , (since ), say , and then", "parent_index": 5, "line_index": 1}, {"bbox": [126, 277, 268, 290], "content": ", so , and", "parent_index": 5, "line_index": 2}, {"bbox": [138, 314, 241, 326], "content": "By our construction", "parent_index": 7, "line_index": 0}, {"bbox": [124, 350, 217, 365], "content": "for .", "parent_index": 9, "line_index": 0}, {"bbox": [138, 366, 255, 378], "content": "By lemma 5.2, part 2),", "parent_index": 10, "line_index": 0}, {"bbox": [125, 402, 296, 417], "content": "for , where .", "parent_index": 12, "line_index": 0}, {"bbox": [137, 416, 486, 432], "content": "By the above calculations the matrices of with", "parent_index": 13, "line_index": 0}, {"bbox": [125, 431, 333, 446], "content": "respect to the basis are", "parent_index": 13, "line_index": 1}, {"bbox": [124, 555, 486, 570], "content": "for , where is the identity matrix, and", "parent_index": 15, "line_index": 0}, {"bbox": [126, 569, 487, 585], "content": ". Since are conjugate in , the are", "parent_index": 15, "line_index": 1}, {"bbox": [126, 584, 388, 598], "content": "all equal, and we have the standard representation.", "parent_index": 15, "line_index": 2}, {"bbox": [137, 612, 485, 626], "content": "Now let us consider when the standard representation is irreducible.", "parent_index": 16, "line_index": 0}, {"bbox": [125, 632, 362, 646], "content": "Lemma 5.3. If then is reducible.", "parent_index": 17, "line_index": 0}, {"bbox": [136, 652, 485, 668], "content": "Proof. If then the vector is a fixed vector.", "parent_index": 18, "line_index": 0}, {"bbox": [125, 687, 370, 702], "content": "Lemma 5.4. If then is irreducible.", "parent_index": 19, "line_index": 0}]
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[]
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[{"bbox": [321, 118, 330, 125], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [437, 114, 486, 127], "content": "I m(A_{i})\\cap", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 174, 140], "content": "I m(A_{i+1})", "parent_index": 0, "subtype": "inline"}, {"bbox": [201, 129, 275, 139], "content": "0\\leq\\,i\\leq n-1", "parent_index": 0, "subtype": "inline"}, {"bbox": [407, 132, 416, 139], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [444, 132, 464, 140], "content": "a_{i+1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [268, 142, 316, 154], "content": "\\{a_{i},a_{i+1}\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [387, 142, 425, 154], "content": "I m(A_{i})", "parent_index": 0, "subtype": "inline"}, {"bbox": [186, 174, 425, 187], "content": "s p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})", "parent_index": 2, "subtype": "interline"}, {"bbox": [220, 194, 235, 205], "content": "B_{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [262, 198, 268, 205], "content": "\\rho", "parent_index": 3, "subtype": "inline"}, {"bbox": [299, 196, 316, 204], "content": "n-", "parent_index": 3, "subtype": "inline"}, {"bbox": [164, 207, 230, 220], "content": "\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [302, 208, 317, 217], "content": "\\mathbb{C}^{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [356, 221, 484, 234], "content": "\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [187, 249, 312, 262], "content": "a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [331, 250, 394, 261], "content": "j\\neq i,\\ \\ i+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [428, 250, 441, 262], "content": "A_{j}", "parent_index": 5, "subtype": "inline"}, {"bbox": [224, 264, 236, 275], "content": "A_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [247, 264, 270, 275], "content": "A_{i+1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [308, 264, 340, 274], "content": "n\\geq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [372, 264, 387, 274], "content": "A_{k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [444, 263, 487, 276], "content": "A_{k}A_{j}=", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 278, 175, 290], "content": "A_{j}A_{k}=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [196, 278, 241, 290], "content": "A_{j}a_{i}=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [239, 295, 371, 309], "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "parent_index": 6, "subtype": "interline"}, {"bbox": [225, 333, 385, 346], "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "parent_index": 8, "subtype": "interline"}, {"bbox": [144, 353, 213, 363], "content": "0\\leq i\\leq n-2", "parent_index": 9, "subtype": "inline"}, {"bbox": [229, 384, 380, 398], "content": "\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},", "parent_index": 11, "subtype": "interline"}, {"bbox": [144, 405, 213, 415], "content": "1\\leq i\\leq n-1", "parent_index": 12, "subtype": "inline"}, {"bbox": [254, 405, 292, 415], "content": "u_{i}\\in\\mathbb{C}^{*}", "parent_index": 12, "subtype": "inline"}, {"bbox": [365, 417, 457, 430], "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "parent_index": 13, "subtype": "inline"}, {"bbox": [229, 435, 311, 444], "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [219, 483, 390, 540], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 14, "subtype": "interline"}, {"bbox": [145, 556, 245, 569], "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "parent_index": 15, "subtype": "inline"}, {"bbox": [289, 556, 300, 568], "content": "I_{k}", "parent_index": 15, "subtype": "inline"}, {"bbox": [340, 557, 372, 567], "content": "k\\,\\times\\,k", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 570, 218, 583], "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}", "parent_index": 15, "subtype": "inline"}, {"bbox": [256, 572, 318, 583], "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "parent_index": 15, "subtype": "inline"}, {"bbox": [411, 572, 425, 582], "content": "B_{n}", "parent_index": 15, "subtype": "inline"}, {"bbox": [454, 575, 464, 582], "content": "u_{i}", "parent_index": 15, "subtype": "inline"}, {"bbox": [212, 632, 242, 644], "content": "u=1", "parent_index": 17, "subtype": "inline"}, {"bbox": [271, 632, 298, 646], "content": "\\tau_{n}(u)", "parent_index": 17, "subtype": "inline"}, {"bbox": [190, 653, 220, 664], "content": "u=1", "parent_index": 18, "subtype": "inline"}, {"bbox": [302, 652, 400, 667], "content": "v=(1,1,1,\\ldots,1)^{T}", "parent_index": 18, "subtype": "inline"}, {"bbox": [212, 687, 242, 701], "content": "u\\ne1", "parent_index": 19, "subtype": "inline"}, {"bbox": [270, 687, 298, 702], "content": "\\tau_{n}(u)", "parent_index": 19, "subtype": "inline"}]
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[]
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Proof. We need to prove that starting from any non-zero vector $x=\textstyle\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\neq0$ . Consider the operator
$$
H=A+A^{2}+A B A=B+B^{2}+B A B,
$$
where $A\,=\,\rho(\sigma_{i-1})$ and $B\;=\;\rho(\sigma_{i})$ . By a direct calculation $H x\;=$ $(u-1)a_{i}e_{i}$ . Because $u\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ .
Now, we have the main result of this paper:
Theorem 5.5 (The Main Theorem). Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\,\geq\,6$ . Let $r\,\geq\,n$ , and let $\rho(\sigma_{1})\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ .
Then $r=n$ and $\rho$ is equivalent to the following representation :
$$
\tau:B_{n}\to G L_{n}(\mathbb{C}),
$$
$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix, and $u\in\mathbb{C}^{*}$ , $u\ne1$ . These representations are non-equivalent for different values of $u$ .
Proof. By Theorem 4.4 the friendship graph of $\rho$ is a chain. Then, by theorem 5.1, $\rho$ is equivalent to a standard representation $\tau(u)$ for some $u\in\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\ne1$ .
Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following
Corollary 5.6. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\ge7$ . Let $c o r a n k(\rho)=2$ .
Then $\rho$ is equivalent to a specialization of the standard representation $\tau_{n}(u)$ , for some $u\neq1,\ u\in\mathbb{C}^{*}$ .
# References
[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.
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<html><body>
<p data-bbox="124 110 487 166">Proof. We need to prove that starting from any non-zero vector $x=\textstyle\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\neq0$ . Consider the operator </p>
<div class="equation" data-bbox="203 173 407 186">$$
H=A+A^{2}+A B A=B+B^{2}+B A B,
$$</div>
<p data-bbox="124 191 486 220">where $A\,=\,\rho(\sigma_{i-1})$ and $B\;=\;\rho(\sigma_{i})$ . By a direct calculation $H x\;=$ $(u-1)a_{i}e_{i}$ . Because $u\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ . </p>
<p data-bbox="136 234 363 248">Now, we have the main result of this paper: </p>
<p data-bbox="125 254 487 297">Theorem 5.5 (The Main Theorem). Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\,\geq\,6$ . Let $r\,\geq\,n$ , and let $\rho(\sigma_{1})\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ . </p>
<p data-bbox="137 297 466 311">Then $r=n$ and $\rho$ is equivalent to the following representation : </p>
<div class="equation" data-bbox="258 317 352 332">$$
\tau:B_{n}\to G L_{n}(\mathbb{C}),
$$</div>
<div class="equation" data-bbox="221 377 389 435">$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="124 448 488 490">for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix, and $u\in\mathbb{C}^{*}$ , $u\ne1$ . These representations are non-equivalent for different values of $u$ . </p>
<p data-bbox="124 497 486 540">Proof. By Theorem 4.4 the friendship graph of $\rho$ is a chain. Then, by theorem 5.1, $\rho$ is equivalent to a standard representation $\tau(u)$ for some $u\in\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\ne1$ . </p>
<p data-bbox="124 540 486 569">Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following </p>
<p data-bbox="124 574 486 603">Corollary 5.6. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\ge7$ . Let $c o r a n k(\rho)=2$ . </p>
<p data-bbox="124 604 486 631">Then $\rho$ is equivalent to a specialization of the standard representation $\tau_{n}(u)$ , for some $u\neq1,\ u\in\mathbb{C}^{*}$ . </p>
<h1 data-bbox="270 643 342 657">References </h1>
<p data-bbox="128 662 486 701">[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658. </p>
</body></html>
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0003047v1
| 14 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. We need to prove that starting from any non-zero vector $x=\\textstyle\\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\\neq0$ . Consider the operator ", "page_idx": 14}, {"type": "equation", "text": "$$\nH=A+A^{2}+A B A=B+B^{2}+B A B,\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "where $A\\,=\\,\\rho(\\sigma_{i-1})$ and $B\\;=\\;\\rho(\\sigma_{i})$ . By a direct calculation $H x\\;=$ $(u-1)a_{i}e_{i}$ . Because $u\\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ . ", "page_idx": 14}, {"type": "text", "text": "Now, we have the main result of this paper: ", "page_idx": 14}, {"type": "text", "text": "Theorem 5.5 (The Main Theorem). Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\,\\geq\\,6$ . Let $r\\,\\geq\\,n$ , and let $\\rho(\\sigma_{1})\\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $r=n$ and $\\rho$ is equivalent to the following representation : ", "page_idx": 14}, {"type": "equation", "text": "$$\n\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix, and $u\\in\\mathbb{C}^{*}$ , $u\\ne1$ . These representations are non-equivalent for different values of $u$ . ", "page_idx": 14}, {"type": "text", "text": "Proof. By Theorem 4.4 the friendship graph of $\\rho$ is a chain. Then, by theorem 5.1, $\\rho$ is equivalent to a standard representation $\\tau(u)$ for some $u\\in\\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\\ne1$ . ", "page_idx": 14}, {"type": "text", "text": "Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following ", "page_idx": 14}, {"type": "text", "text": "Corollary 5.6. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\ge7$ . Let $c o r a n k(\\rho)=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $\\rho$ is equivalent to a specialization of the standard representation $\\tau_{n}(u)$ , for some $u\\neq1,\\ u\\in\\mathbb{C}^{*}$ . ", "page_idx": 14}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658. ", "page_idx": 14}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 110, 487, 166], "lines": [{"bbox": [137, 113, 485, 126], "spans": [{"bbox": [137, 113, 485, 126], "score": 1.0, "content": "Proof. We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "type": "text"}], "index": 27}, {"bbox": [146, 678, 332, 689], "spans": [{"bbox": [146, 678, 332, 689], "score": 1.0, "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "type": "text"}], "index": 28}, {"bbox": [131, 690, 484, 701], "spans": [{"bbox": [131, 690, 484, 701], "score": 1.0, "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "type": "text"}], "index": 29}], "index": 28}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 90, 389, 101], "lines": [{"bbox": [223, 93, 389, 102], "spans": [{"bbox": [223, 93, 389, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [475, 91, 486, 100], "lines": [{"bbox": [473, 93, 486, 102], "spans": [{"bbox": [473, 93, 486, 102], "score": 1.0, "content": "15", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 166], "lines": [{"bbox": [137, 113, 485, 126], "spans": [{"bbox": [137, 113, 485, 126], "score": 1.0, "content": "Proof. We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [125, 113, 486, 168]}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [126, 194, 486, 222]}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [137, 235, 363, 250]}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [125, 257, 487, 299]}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [140, 299, 465, 313]}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15, "bbox_fs": [125, 451, 487, 492]}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [126, 500, 485, 542]}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [126, 542, 486, 571]}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [126, 577, 487, 605]}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [126, 605, 487, 632]}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "type": "text"}], "index": 27}, {"bbox": [146, 678, 332, 689], "spans": [{"bbox": [146, 678, 332, 689], "score": 1.0, "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "type": "text"}], "index": 28}, {"bbox": [131, 690, 484, 701], "spans": [{"bbox": [131, 690, 484, 701], "score": 1.0, "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "type": "text"}], "index": 29}], "index": 28, "bbox_fs": [131, 665, 486, 701]}]}
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[{"type": "text", "bbox": [124, 110, 487, 166], "content": "Proof. We need to prove that starting from any non-zero vector , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors . To do this, take such that . Consider the operator", "index": 0}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "content": "", "index": 1}, {"type": "text", "bbox": [124, 191, 486, 220], "content": "where and . By a direct calculation . Because the vector is a non-zero multiple of .", "index": 2}, {"type": "text", "bbox": [136, 234, 363, 248], "content": "Now, we have the main result of this paper:", "index": 3}, {"type": "text", "bbox": [125, 254, 487, 297], "content": "Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .", "index": 4}, {"type": "text", "bbox": [137, 297, 466, 311], "content": "Then and is equivalent to the following representation :", "index": 5}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "content": "", "index": 6}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "content": "", "index": 7}, {"type": "text", "bbox": [124, 448, 488, 490], "content": "for , where is the identity matrix, and , . These representations are non-equivalent for different values of .", "index": 8}, {"type": "text", "bbox": [124, 497, 486, 540], "content": "Proof. By Theorem 4.4 the friendship graph of is a chain. Then, by theorem 5.1, is equivalent to a standard representation for some . By Lemmas 5.3 and 5.4 .", "index": 9}, {"type": "text", "bbox": [124, 540, 486, 569], "content": "Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following", "index": 10}, {"type": "text", "bbox": [124, 574, 486, 603], "content": "Corollary 5.6. Let be an irreducible representation of for . Let .", "index": 11}, {"type": "text", "bbox": [124, 604, 486, 631], "content": "Then is equivalent to a specialization of the standard representation , for some .", "index": 12}, {"type": "title", "bbox": [270, 643, 342, 657], "content": "References", "index": 13}, {"type": "text", "bbox": [128, 662, 486, 701], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "index": 14}]
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[{"bbox": [137, 113, 485, 126], "content": "Proof. We need to prove that starting from any non-zero vector", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 484, 140], "content": ", we can generate the whole space. Obviously, it is enough", "parent_index": 0, "line_index": 1}, {"bbox": [125, 141, 486, 154], "content": "to show that we can generate one of the standard basis vectors . To", "parent_index": 0, "line_index": 2}, {"bbox": [125, 155, 405, 168], "content": "do this, take such that . Consider the operator", "parent_index": 0, "line_index": 3}, {"bbox": [126, 194, 486, 208], "content": "where and . By a direct calculation", "parent_index": 2, "line_index": 0}, {"bbox": [126, 207, 484, 222], "content": ". Because the vector is a non-zero multiple of .", "parent_index": 2, "line_index": 1}, {"bbox": [137, 235, 363, 250], "content": "Now, we have the main result of this paper:", "parent_index": 3, "line_index": 0}, {"bbox": [125, 257, 486, 272], "content": "Theorem 5.5 (The Main Theorem). Let be an ir-", "parent_index": 4, "line_index": 0}, {"bbox": [126, 271, 487, 285], "content": "reducible representation of for . Let , and let", "parent_index": 4, "line_index": 1}, {"bbox": [126, 285, 263, 299], "content": "with .", "parent_index": 4, "line_index": 2}, {"bbox": [140, 299, 465, 313], "content": "Then and is equivalent to the following representation :", "parent_index": 5, "line_index": 0}, {"bbox": [125, 451, 484, 464], "content": "for , where is the identity matrix, and ,", "parent_index": 8, "line_index": 0}, {"bbox": [126, 465, 487, 479], "content": ". These representations are non-equivalent for different values of", "parent_index": 8, "line_index": 1}, {"bbox": [126, 481, 137, 492], "content": ".", "parent_index": 8, "line_index": 2}, {"bbox": [137, 500, 485, 514], "content": "Proof. By Theorem 4.4 the friendship graph of is a chain. Then,", "parent_index": 9, "line_index": 0}, {"bbox": [126, 514, 485, 528], "content": "by theorem 5.1, is equivalent to a standard representation for", "parent_index": 9, "line_index": 1}, {"bbox": [126, 528, 355, 542], "content": "some . By Lemmas 5.3 and 5.4 .", "parent_index": 9, "line_index": 2}, {"bbox": [137, 542, 486, 555], "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "parent_index": 10, "line_index": 0}, {"bbox": [126, 554, 346, 571], "content": "(see [3], Theorem 23), we get the following", "parent_index": 10, "line_index": 1}, {"bbox": [126, 577, 487, 592], "content": "Corollary 5.6. Let be an irreducible representation", "parent_index": 11, "line_index": 0}, {"bbox": [127, 591, 310, 605], "content": "of for . Let .", "parent_index": 11, "line_index": 1}, {"bbox": [139, 605, 487, 619], "content": "Then is equivalent to a specialization of the standard representation", "parent_index": 12, "line_index": 0}, {"bbox": [126, 619, 287, 632], "content": ", for some .", "parent_index": 12, "line_index": 1}, {"bbox": [270, 645, 342, 658], "content": "References", "parent_index": 13, "line_index": 0}, {"bbox": [131, 665, 486, 678], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "parent_index": 14, "line_index": 0}, {"bbox": [146, 678, 332, 689], "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "parent_index": 14, "line_index": 1}, {"bbox": [131, 690, 484, 701], "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "parent_index": 14, "line_index": 2}]
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[{"bbox": [126, 128, 181, 140], "content": "x=\\textstyle\\sum a_{i}e_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [454, 146, 462, 153], "content": "e_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [194, 157, 198, 165], "content": "i", "parent_index": 0, "subtype": "inline"}, {"bbox": [253, 156, 285, 167], "content": "a_{i}\\neq0", "parent_index": 0, "subtype": "inline"}, {"bbox": [203, 173, 407, 186], "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "parent_index": 1, "subtype": "interline"}, {"bbox": [161, 195, 227, 207], "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [257, 195, 313, 208], "content": "B\\;=\\;\\rho(\\sigma_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [452, 195, 486, 207], "content": "H x\\;=", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 209, 181, 221], "content": "(u-1)a_{i}e_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 209, 262, 221], "content": "u\\ne1", "parent_index": 2, "subtype": "inline"}, {"bbox": [321, 209, 339, 218], "content": "H x", "parent_index": 2, "subtype": "inline"}, {"bbox": [470, 212, 479, 220], "content": "e_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [343, 258, 435, 270], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 4, "subtype": "inline"}, {"bbox": [266, 271, 281, 284], "content": "B_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [304, 271, 337, 284], "content": "n\\,\\geq\\,6", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 272, 396, 284], "content": "r\\,\\geq\\,n", "parent_index": 4, "subtype": "inline"}, {"bbox": [444, 272, 487, 285], "content": "\\rho(\\sigma_{1})\\,=", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 287, 160, 297], "content": "1+A_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [190, 286, 259, 299], "content": "r a n k(A_{1})=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [168, 304, 197, 309], "content": "r=n", "parent_index": 5, "subtype": "inline"}, {"bbox": [223, 303, 230, 312], "content": "\\rho", "parent_index": 5, "subtype": "inline"}, {"bbox": [258, 317, 352, 332], "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "parent_index": 6, "subtype": "interline"}, {"bbox": [221, 377, 389, 435], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 7, "subtype": "interline"}, {"bbox": [144, 452, 231, 464], "content": "i=1,2,\\dots,n-1", "parent_index": 8, "subtype": "inline"}, {"bbox": [270, 452, 281, 463], "content": "I_{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [315, 452, 339, 462], "content": "k\\times k", "parent_index": 8, "subtype": "inline"}, {"bbox": [447, 452, 482, 462], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 466, 154, 478], "content": "u\\ne1", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 483, 133, 489], "content": "u", "parent_index": 8, "subtype": "inline"}, {"bbox": [388, 505, 394, 513], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [212, 519, 218, 527], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [444, 515, 466, 528], "content": "\\tau(u)", "parent_index": 9, "subtype": "inline"}, {"bbox": [155, 530, 190, 539], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 9, "subtype": "inline"}, {"bbox": [322, 529, 351, 541], "content": "u\\ne1", "parent_index": 9, "subtype": "inline"}, {"bbox": [232, 578, 321, 590], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 11, "subtype": "inline"}, {"bbox": [139, 591, 155, 604], "content": "B_{n}", "parent_index": 11, "subtype": "inline"}, {"bbox": [177, 591, 207, 603], "content": "n\\ge7", "parent_index": 11, "subtype": "inline"}, {"bbox": [233, 592, 307, 605], "content": "c o r a n k(\\rho)=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [167, 607, 174, 618], "content": "\\rho", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 619, 153, 632], "content": "\\tau_{n}(u)", "parent_index": 12, "subtype": "inline"}, {"bbox": [208, 619, 283, 632], "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "parent_index": 12, "subtype": "inline"}]
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[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322.
[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388.
[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999.
[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]
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<p data-bbox="130 112 487 208">[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322.
[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388.
[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999.
[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486. </p>
<p data-bbox="125 218 484 255">Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected] </p>
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# IMAGINARY QUADRATIC FIELDS $k$ WITH $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ AND RANK $\mathrm{Cl}_{2}(k^{1})=2$
E. BENJAMIN, F. LEMMERMEYER, C. SNYDER
Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ such that $\mathrm{Cl}_{2}(k^{1})$ has 2 generators.
# 1. Introduction
Let $k$ be an algebraic number field with $\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\,=\,k$ and kn+1 = (kn)1. Then
$$
k^{0}\subseteq k^{1}\subseteq k^{2}\subseteq\cdots\subseteq k^{n}\subseteq\cdot\cdot.
$$
is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length.
At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\,\geq\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\mathrm{Cl}_{2}(k^{1})\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.
From a classification of imaginary quadratic number fields $k$ with $\mathrm{Cl_{2}}(k)\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\left(d_{i}/p_{j}\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases:
1991 Mathematics Subject Classification. Primary 11R37.
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<h1 data-bbox="129 142 482 171">IMAGINARY QUADRATIC FIELDS $k$ WITH $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ AND RANK $\mathrm{Cl}_{2}(k^{1})=2$ </h1>
<p data-bbox="205 189 406 199">E. BENJAMIN, F. LEMMERMEYER, C. SNYDER </p>
<p data-bbox="160 218 450 248">Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ such that $\mathrm{Cl}_{2}(k^{1})$ has 2 generators. </p>
<h1 data-bbox="265 282 346 294">1. Introduction </h1>
<p data-bbox="125 300 485 348">Let $k$ be an algebraic number field with $\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\,=\,k$ and kn+1 = (kn)1. Then </p>
<div class="equation" data-bbox="240 354 371 365">$$
k^{0}\subseteq k^{1}\subseteq k^{2}\subseteq\cdots\subseteq k^{n}\subseteq\cdot\cdot.
$$</div>
<p data-bbox="125 368 486 404">is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length. </p>
<p data-bbox="126 405 486 607">At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\,\geq\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\mathrm{Cl}_{2}(k^{1})\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. </p>
<p data-bbox="126 608 486 667">From a classification of imaginary quadratic number fields $k$ with $\mathrm{Cl_{2}}(k)\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\left(d_{i}/p_{j}\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases: </p>
<p data-bbox="137 689 353 700">1991 Mathematics Subject Classification. Primary 11R37. </p>
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[{"type": "text", "text": "IMAGINARY QUADRATIC FIELDS $k$ WITH $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ AND RANK $\\mathrm{Cl}_{2}(k^{1})=2$ ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER ", "page_idx": 0}, {"type": "text", "text": "Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ such that $\\mathrm{Cl}_{2}(k^{1})$ has 2 generators. ", "page_idx": 0}, {"type": "text", "text": "1. Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Let $k$ be an algebraic number field with $\\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\\,=\\,k$ and kn+1 = (kn)1. Then ", "page_idx": 0}, {"type": "equation", "text": "$$\nk^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length. ", "page_idx": 0}, {"type": "text", "text": "At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\\mathrm{Cl}_{2}(k^{1})\\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. ", "page_idx": 0}, {"type": "text", "text": "From a classification of imaginary quadratic number fields $k$ with $\\mathrm{Cl_{2}}(k)\\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\\left(d_{i}/p_{j}\\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases: ", "page_idx": 0}, {"type": "text", "text": "1991 Mathematics Subject Classification. Primary 11R37. ", "page_idx": 0}]
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Denote by ", "type": "text"}, {"bbox": [260, 315, 270, 323], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [270, 314, 390, 327], "score": 1.0, "content": " the Hilbert 2-class field of ", "type": "text"}, {"bbox": [390, 316, 396, 323], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [396, 314, 486, 327], "score": 1.0, "content": " (in the wide sense).", "type": "text"}], "index": 8}, {"bbox": [126, 325, 487, 339], "spans": [{"bbox": [126, 325, 164, 339], "score": 1.0, "content": "Also let ", "type": "text"}, {"bbox": [164, 328, 176, 335], "score": 0.9, "content": "k^{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 325, 199, 339], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [200, 330, 206, 335], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [206, 325, 432, 339], "score": 1.0, "content": " a nonnegative integer) be defined inductively as: ", "type": "text"}, {"bbox": [432, 327, 465, 335], "score": 0.93, "content": "k^{0}\\,=\\,k", "type": "inline_equation", "height": 8, "width": 33}, {"bbox": [465, 325, 487, 339], "score": 1.0, "content": " and", "type": "text"}], "index": 9}, {"bbox": [125, 336, 215, 351], "spans": [{"bbox": [125, 336, 215, 351], "score": 1.0, "content": "kn+1 = (kn)1. Then", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 368, 486, 404], "lines": [{"bbox": [124, 369, 484, 383], "spans": [{"bbox": [124, 369, 268, 383], "score": 1.0, "content": "is called the 2-class field tower of ", "type": "text"}, {"bbox": [268, 372, 274, 379], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [274, 369, 289, 383], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [289, 375, 296, 379], "score": 0.89, "content": "n", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [296, 369, 436, 383], "score": 1.0, "content": " is the minimal integer such that", "type": "text"}, {"bbox": [437, 371, 482, 379], "score": 0.92, "content": "k^{n}=k^{n+1}", "type": "inline_equation", "height": 8, "width": 45}, {"bbox": [482, 369, 484, 383], "score": 1.0, "content": ",", "type": "text"}], "index": 12}, {"bbox": [126, 383, 486, 394], "spans": [{"bbox": [126, 383, 148, 394], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [149, 387, 155, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [155, 383, 350, 394], "score": 1.0, "content": " is called the length of the tower. If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. 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We give a characterization of those ", "type": "text"}, {"bbox": [361, 231, 365, 237], "score": 0.73, "content": "k", "type": "inline_equation", "height": 6, "width": 4}, {"bbox": [366, 230, 387, 240], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [388, 231, 449, 239], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 8, "width": 61}], "index": 4}, {"bbox": [161, 240, 294, 250], "spans": [{"bbox": [161, 240, 199, 250], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [199, 240, 227, 249], "score": 0.87, "content": "\\mathrm{Cl}_{2}(k^{1})", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [227, 240, 294, 250], "score": 1.0, "content": " has 2 generators.", "type": "text"}], "index": 5}], "index": 4, "bbox_fs": [161, 220, 450, 250]}, {"type": "title", "bbox": [265, 282, 346, 294], "lines": [{"bbox": [264, 284, 348, 296], "spans": [{"bbox": [264, 284, 348, 296], "score": 1.0, "content": "1. Introduction", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [125, 300, 485, 348], "lines": [{"bbox": [137, 302, 486, 315], "spans": [{"bbox": [137, 302, 155, 315], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [155, 304, 161, 311], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 302, 310, 315], "score": 1.0, "content": " be an algebraic number field with ", "type": "text"}, {"bbox": [310, 303, 338, 314], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [339, 302, 486, 315], "score": 1.0, "content": ", the Sylow 2-subgroup of its ideal", "type": "text"}], "index": 7}, {"bbox": [124, 314, 486, 327], "spans": [{"bbox": [124, 314, 180, 327], "score": 1.0, "content": "class group, ", "type": "text"}, {"bbox": [181, 315, 204, 326], "score": 0.85, "content": "\\operatorname{Cl}(k)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [204, 314, 259, 327], "score": 1.0, "content": ". Denote by ", "type": "text"}, {"bbox": [260, 315, 270, 323], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [270, 314, 390, 327], "score": 1.0, "content": " the Hilbert 2-class field of ", "type": "text"}, {"bbox": [390, 316, 396, 323], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [396, 314, 486, 327], "score": 1.0, "content": " (in the wide sense).", "type": "text"}], "index": 8}, {"bbox": [126, 325, 487, 339], "spans": [{"bbox": [126, 325, 164, 339], "score": 1.0, "content": "Also let ", "type": "text"}, {"bbox": [164, 328, 176, 335], "score": 0.9, "content": "k^{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 325, 199, 339], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [200, 330, 206, 335], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [206, 325, 432, 339], "score": 1.0, "content": " a nonnegative integer) be defined inductively as: ", "type": "text"}, {"bbox": [432, 327, 465, 335], "score": 0.93, "content": "k^{0}\\,=\\,k", "type": "inline_equation", "height": 8, "width": 33}, {"bbox": [465, 325, 487, 339], "score": 1.0, "content": " and", "type": "text"}], "index": 9}, {"bbox": [125, 336, 215, 351], "spans": [{"bbox": [125, 336, 215, 351], "score": 1.0, "content": "kn+1 = (kn)1. Then", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [124, 302, 487, 351]}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 368, 486, 404], "lines": [{"bbox": [124, 369, 484, 383], "spans": [{"bbox": [124, 369, 268, 383], "score": 1.0, "content": "is called the 2-class field tower of ", "type": "text"}, {"bbox": [268, 372, 274, 379], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [274, 369, 289, 383], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [289, 375, 296, 379], "score": 0.89, "content": "n", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [296, 369, 436, 383], "score": 1.0, "content": " is the minimal integer such that", "type": "text"}, {"bbox": [437, 371, 482, 379], "score": 0.92, "content": "k^{n}=k^{n+1}", "type": "inline_equation", "height": 8, "width": 45}, {"bbox": [482, 369, 484, 383], "score": 1.0, "content": ",", "type": "text"}], "index": 12}, {"bbox": [126, 383, 486, 394], "spans": [{"bbox": [126, 383, 148, 394], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [149, 387, 155, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [155, 383, 350, 394], "score": 1.0, "content": " is called the length of the tower. If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13, "bbox_fs": [124, 369, 486, 406]}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23, "bbox_fs": [124, 407, 487, 610]}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34, "bbox_fs": [125, 608, 487, 670]}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. Primary 11R37.", "type": "text"}], "index": 37}], "index": 37, "bbox_fs": [138, 691, 353, 701]}]}
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[{"type": "title", "bbox": [129, 142, 482, 171], "content": "IMAGINARY QUADRATIC FIELDS WITH AND RANK", "index": 0}, {"type": "text", "bbox": [205, 189, 406, 199], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "index": 1}, {"type": "text", "bbox": [160, 218, 450, 248], "content": "Abstract. Let be an imaginary quadratic number field and the Hilbert 2-class field of . We give a characterization of those with such that has 2 generators.", "index": 2}, {"type": "title", "bbox": [265, 282, 346, 294], "content": "1. Introduction", "index": 3}, {"type": "text", "bbox": [125, 300, 485, 348], "content": "Let be an algebraic number field with , the Sylow 2-subgroup of its ideal class group, . Denote by the Hilbert 2-class field of (in the wide sense). Also let (for a nonnegative integer) be defined inductively as: and kn+1 = (kn)1. Then", "index": 4}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "content": "", "index": 5}, {"type": "text", "bbox": [125, 368, 486, 404], "content": "is called the 2-class field tower of . If is the minimal integer such that , then is called the length of the tower. If no such exists, then the tower is said to be of infinite length.", "index": 6}, {"type": "text", "bbox": [126, 405, 486, 607], "content": "At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field is infinite. However, it is known by group theoretic results (see [2]) that if r , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank . We have already started this program for imaginary quadratic number fields . In [1] we classified all imaginary quadratic fields whose 2-class field has cyclic 2-class group. In this paper we determine when has rank 2 for imaginary quadratic fields with of type . (The notation means the direct sum of a group of order 2 and a cyclic group of order .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.", "index": 7}, {"type": "text", "bbox": [126, 608, 486, 667], "content": "From a classification of imaginary quadratic number fields with and our results from [1] we see that it suffices to consider discriminants with prime discriminants , such that exactly one of the equals (we let denote the prime dividing ); thus there are only two cases:", "index": 8}, {"type": "text", "bbox": [137, 689, 353, 700], "content": "1991 Mathematics Subject Classification. Primary 11R37.", "index": 9}]
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[{"bbox": [130, 145, 480, 158], "content": "IMAGINARY QUADRATIC FIELDS WITH AND", "parent_index": 0, "line_index": 0}, {"bbox": [259, 159, 351, 172], "content": "RANK", "parent_index": 0, "line_index": 1}, {"bbox": [205, 192, 406, 201], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "parent_index": 1, "line_index": 0}, {"bbox": [162, 220, 450, 230], "content": "Abstract. Let be an imaginary quadratic number field and the Hilbert", "parent_index": 2, "line_index": 0}, {"bbox": [161, 230, 449, 240], "content": "2-class field of . We give a characterization of those with", "parent_index": 2, "line_index": 1}, {"bbox": [161, 240, 294, 250], "content": "such that has 2 generators.", "parent_index": 2, "line_index": 2}, {"bbox": [264, 284, 348, 296], "content": "1. Introduction", "parent_index": 3, "line_index": 0}, {"bbox": [137, 302, 486, 315], "content": "Let be an algebraic number field with , the Sylow 2-subgroup of its ideal", "parent_index": 4, "line_index": 0}, {"bbox": [124, 314, 486, 327], "content": "class group, . Denote by the Hilbert 2-class field of (in the wide sense).", "parent_index": 4, "line_index": 1}, {"bbox": [126, 325, 487, 339], "content": "Also let (for a nonnegative integer) be defined inductively as: and", "parent_index": 4, "line_index": 2}, {"bbox": [125, 336, 215, 351], "content": "kn+1 = (kn)1. Then", "parent_index": 4, "line_index": 3}, {"bbox": [124, 369, 484, 383], "content": "is called the 2-class field tower of . If is the minimal integer such that ,", "parent_index": 6, "line_index": 0}, {"bbox": [126, 383, 486, 394], "content": "then is called the length of the tower. If no such exists, then the tower is said", "parent_index": 6, "line_index": 1}, {"bbox": [126, 395, 226, 406], "content": "to be of infinite length.", "parent_index": 6, "line_index": 2}, {"bbox": [138, 407, 485, 417], "content": "At present there is no known decision procedure to determine whether or not", "parent_index": 7, "line_index": 0}, {"bbox": [126, 419, 485, 429], "content": "the (2-)class field tower of a given field is infinite. However, it is known by", "parent_index": 7, "line_index": 1}, {"bbox": [125, 430, 486, 442], "content": "group theoretic results (see [2]) that if r , then the tower is finite,", "parent_index": 7, "line_index": 2}, {"bbox": [125, 442, 486, 454], "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "parent_index": 7, "line_index": 3}, {"bbox": [125, 454, 487, 467], "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "parent_index": 7, "line_index": 4}, {"bbox": [124, 465, 487, 479], "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "parent_index": 7, "line_index": 5}, {"bbox": [125, 478, 486, 490], "content": "fields with rank (let us mention in particular Schmithals [13]) have", "parent_index": 7, "line_index": 6}, {"bbox": [125, 490, 485, 502], "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "parent_index": 7, "line_index": 7}, {"bbox": [124, 502, 487, 515], "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "parent_index": 7, "line_index": 8}, {"bbox": [125, 514, 487, 526], "content": "good starting point would be to find a procedure for sieving out those fields with", "parent_index": 7, "line_index": 9}, {"bbox": [124, 525, 487, 539], "content": "rank . We have already started this program for imaginary quadratic", "parent_index": 7, "line_index": 10}, {"bbox": [126, 538, 486, 550], "content": "number fields . In [1] we classified all imaginary quadratic fields whose 2-class field", "parent_index": 7, "line_index": 11}, {"bbox": [126, 548, 487, 563], "content": "has cyclic 2-class group. In this paper we determine when has rank 2", "parent_index": 7, "line_index": 12}, {"bbox": [125, 562, 485, 574], "content": "for imaginary quadratic fields with of type . (The notation", "parent_index": 7, "line_index": 13}, {"bbox": [125, 574, 487, 586], "content": "means the direct sum of a group of order 2 and a cyclic group of order .) The", "parent_index": 7, "line_index": 14}, {"bbox": [125, 586, 486, 597], "content": "group theoretic results mentioned above also show that such fields have 2-class field", "parent_index": 7, "line_index": 15}, {"bbox": [125, 598, 204, 610], "content": "tower of length 2.", "parent_index": 7, "line_index": 16}, {"bbox": [136, 608, 486, 623], "content": "From a classification of imaginary quadratic number fields with", "parent_index": 8, "line_index": 0}, {"bbox": [126, 622, 487, 633], "content": "and our results from [1] we see that it suffices to consider discriminants", "parent_index": 8, "line_index": 1}, {"bbox": [126, 633, 487, 645], "content": "with prime discriminants , such that exactly one of", "parent_index": 8, "line_index": 2}, {"bbox": [125, 645, 485, 658], "content": "the equals (we let denote the prime dividing ); thus there are only", "parent_index": 8, "line_index": 3}, {"bbox": [125, 658, 171, 670], "content": "two cases:", "parent_index": 8, "line_index": 4}, {"bbox": [138, 691, 353, 701], "content": "1991 Mathematics Subject Classification. Primary 11R37.", "parent_index": 9, "line_index": 0}]
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[{"bbox": [329, 148, 335, 155], "content": "k", "parent_index": 0, "subtype": "inline"}, {"bbox": [378, 147, 448, 158], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 0, "subtype": "inline"}, {"bbox": [300, 160, 351, 172], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [223, 222, 227, 227], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [396, 220, 405, 227], "content": "k^{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 231, 222, 237], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [361, 231, 365, 237], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [388, 231, 449, 239], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [199, 240, 227, 249], "content": "\\mathrm{Cl}_{2}(k^{1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [155, 304, 161, 311], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [310, 303, 338, 314], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [181, 315, 204, 326], "content": "\\operatorname{Cl}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [260, 315, 270, 323], "content": "k^{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [390, 316, 396, 323], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [164, 328, 176, 335], "content": "k^{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [200, 330, 206, 335], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [432, 327, 465, 335], "content": "k^{0}\\,=\\,k", "parent_index": 4, "subtype": "inline"}, {"bbox": [240, 354, 371, 365], "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "parent_index": 5, "subtype": "interline"}, {"bbox": [268, 372, 274, 379], "content": "k", "parent_index": 6, "subtype": "inline"}, {"bbox": [289, 375, 296, 379], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [437, 371, 482, 379], "content": "k^{n}=k^{n+1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [149, 387, 155, 392], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [350, 387, 357, 392], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [310, 420, 316, 427], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [303, 430, 374, 442], "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "parent_index": 7, "subtype": "inline"}, {"bbox": [198, 479, 252, 489], "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "parent_index": 7, "subtype": "inline"}, {"bbox": [147, 526, 200, 537], "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "parent_index": 7, "subtype": "inline"}, {"bbox": [186, 540, 192, 547], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 550, 136, 559], "content": "k^{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [402, 550, 434, 561], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [258, 564, 263, 571], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [289, 563, 317, 573], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 7, "subtype": "inline"}, {"bbox": [354, 563, 384, 573], "content": "(2,2^{m})", "parent_index": 7, "subtype": "inline"}, {"bbox": [455, 563, 485, 573], "content": "(2,2^{m})", "parent_index": 7, "subtype": "inline"}, {"bbox": [443, 576, 456, 583], "content": "2^{m}", "parent_index": 7, "subtype": "inline"}, {"bbox": [409, 611, 415, 618], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [444, 610, 486, 621], "content": "\\mathrm{Cl_{2}}(k)\\simeq", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 623, 156, 633], "content": "(2,2^{m})", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 635, 175, 644], "content": "d=d_{1}d_{2}d_{3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [293, 635, 337, 644], "content": "d_{1},d_{2}>0", "parent_index": 8, "subtype": "inline"}, {"bbox": [344, 635, 374, 644], "content": "d_{3}<0", "parent_index": 8, "subtype": "inline"}, {"bbox": [143, 646, 173, 657], "content": "\\left(d_{i}/p_{j}\\right)", "parent_index": 8, "subtype": "inline"}, {"bbox": [207, 648, 220, 655], "content": "^{-1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [255, 650, 264, 657], "content": "p_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [383, 647, 393, 657], "content": "d_{j}", "parent_index": 8, "subtype": "inline"}]
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The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\mathrm{Cl_{2}}(k)$ is $d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work.
The main result of the paper is that rank $\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following
Theorem 1. Let $k$ be a complex quadratic number field with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\,>\,0$ and $-4\,\ne\,d_{3}\,<\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$ .
This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\mathrm{Cl}_{2}(k^{1})\,=\,2$ ; it remains to solve these problems for fields with rank $\mathrm{Cl}_{2}(k)=3$ and those with $\mathrm{Cl}_{2}(k)\supseteq(4,4)$ since we know that rank $\mathrm{Cl}_{2}(k^{1})\geq5$ whenever rank $\mathrm{Cl}_{2}(k)\geq4$ (using Schur multipliers as in [1]).
As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$ and noncyclic $\mathrm{Cl_{2}}(k^{1})$ .
Table 1
<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>
Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\prime}\simeq(2,4)$ satisfy $\mathrm{rank}\,G^{\prime}\leq3$ ), see [3].
In order to verify that $\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$ it is sufficient to show that its genus class field $k_{\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$ , and the claim follows.
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<p data-bbox="125 127 486 186">The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\mathrm{Cl_{2}}(k)$ is $d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. </p>
<p data-bbox="125 187 487 211">The main result of the paper is that rank $\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following </p>
<p data-bbox="125 217 487 290">Theorem 1. Let $k$ be a complex quadratic number field with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\,>\,0$ and $-4\,\ne\,d_{3}\,<\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$ . </p>
<p data-bbox="125 296 486 344">This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\mathrm{Cl}_{2}(k^{1})\,=\,2$ ; it remains to solve these problems for fields with rank $\mathrm{Cl}_{2}(k)=3$ and those with $\mathrm{Cl}_{2}(k)\supseteq(4,4)$ since we know that rank $\mathrm{Cl}_{2}(k^{1})\geq5$ whenever rank $\mathrm{Cl}_{2}(k)\geq4$ (using Schur multipliers as in [1]). </p>
<p data-bbox="124 344 486 380">As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$ and noncyclic $\mathrm{Cl_{2}}(k^{1})$ . </p>
<div class="table" data-bbox="126 429 489 588"><p class="caption" data-bbox="285 392 325 403">Table 1 </p><table data-bbox="126 429 489 588"><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></div>
<p data-bbox="125 603 486 651">Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\prime}\simeq(2,4)$ satisfy $\mathrm{rank}\,G^{\prime}\leq3$ ), see [3]. </p>
<p data-bbox="124 651 486 699">In order to verify that $\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$ it is sufficient to show that its genus class field $k_{\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$ , and the claim follows. </p>
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0003244v1
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[{"type": "text", "text": "The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\\mathrm{Cl_{2}}(k)$ is $d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. ", "page_idx": 1}, {"type": "text", "text": "The main result of the paper is that rank $\\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following ", "page_idx": 1}, {"type": "text", "text": "Theorem 1. Let $k$ be a complex quadratic number field with $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\\,>\\,0$ and $-4\\,\\ne\\,d_{3}\\,<\\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K$ . ", "page_idx": 1}, {"type": "text", "text": "This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\\mathrm{Cl}_{2}(k^{1})\\,=\\,2$ ; it remains to solve these problems for fields with rank $\\mathrm{Cl}_{2}(k)=3$ and those with $\\mathrm{Cl}_{2}(k)\\supseteq(4,4)$ since we know that rank $\\mathrm{Cl}_{2}(k^{1})\\geq5$ whenever rank $\\mathrm{Cl}_{2}(k)\\geq4$ (using Schur multipliers as in [1]). ", "page_idx": 1}, {"type": "text", "text": "As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2$ and noncyclic $\\mathrm{Cl_{2}}(k^{1})$ . ", "page_idx": 1}, {"type": "table", "img_path": "images/582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg", "table_caption": ["Table 1 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>\n\n", "page_idx": 1}, {"type": "text", "text": "Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\\prime}\\simeq(2,4)$ satisfy $\\mathrm{rank}\\,G^{\\prime}\\leq3$ ), see [3]. ", "page_idx": 1}, {"type": "text", "text": "In order to verify that $\\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)$ it is sufficient to show that its genus class field $k_{\\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})$ , and the claim follows. ", "page_idx": 1}]
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B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [347, 1811, 1351, 1811, 1351, 1943, 347, 1943], "score": 0.956}, {"category_id": 1, "poly": [348, 521, 1353, 521, 1353, 588, 348, 588], "score": 0.893}, {"category_id": 6, "poly": [793, 1089, 904, 1089, 904, 1122, 793, 1122], "score": 0.862}, {"category_id": 2, "poly": [348, 253, 366, 253, 366, 275, 348, 275], "score": 0.783}, {"category_id": 8, "poly": [367, 311, 895, 311, 895, 349, 367, 349], "score": 0.653}, {"category_id": 1, "poly": [349, 521, 1353, 521, 1353, 588, 349, 588], "score": 0.383}, {"category_id": 13, "poly": [351, 778, 539, 778, 539, 810, 351, 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{"preproc_blocks": [{"type": "text", "bbox": [125, 127, 486, 186], "lines": [{"bbox": [126, 128, 484, 141], "spans": [{"bbox": [126, 128, 145, 141], "score": 1.0, "content": "The", "type": "text"}, {"bbox": [146, 131, 157, 140], "score": 0.93, "content": "C_{4}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [158, 128, 393, 141], "score": 1.0, "content": "-factorization corresponding to the nontrivial 4-part of ", "type": "text"}, {"bbox": [393, 130, 421, 141], "score": 0.89, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [421, 128, 433, 141], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [433, 131, 484, 140], "score": 0.93, "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "type": "inline_equation", "height": 9, "width": 51}], "index": 0}, {"bbox": [126, 141, 486, 153], "spans": [{"bbox": [126, 141, 195, 153], "score": 1.0, "content": "in case A) and ", "type": "text"}, {"bbox": [195, 143, 252, 151], "score": 0.94, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 57}, {"bbox": [252, 141, 486, 153], "score": 1.0, "content": " in case B). Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 102], "spans": [{"bbox": [126, 93, 131, 102], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 127, 486, 186], "lines": [{"bbox": [126, 128, 484, 141], "spans": [{"bbox": [126, 128, 145, 141], "score": 1.0, "content": "The", "type": "text"}, {"bbox": [146, 131, 157, 140], "score": 0.93, "content": "C_{4}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [158, 128, 393, 141], "score": 1.0, "content": "-factorization corresponding to the nontrivial 4-part of ", "type": "text"}, {"bbox": [393, 130, 421, 141], "score": 0.89, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [421, 128, 433, 141], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [433, 131, 484, 140], "score": 0.93, "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "type": "inline_equation", "height": 9, "width": 51}], "index": 0}, {"bbox": [126, 141, 486, 153], "spans": [{"bbox": [126, 141, 195, 153], "score": 1.0, "content": "in case A) and ", "type": "text"}, {"bbox": [195, 143, 252, 151], "score": 0.94, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 57}, {"bbox": [252, 141, 486, 153], "score": 1.0, "content": " in case B). Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [126, 128, 486, 188]}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [125, 188, 488, 214]}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5, "bbox_fs": [125, 219, 487, 291]}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5, "bbox_fs": [126, 298, 486, 346]}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [125, 346, 487, 382]}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5, "bbox_fs": [126, 605, 486, 653]}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [125, 652, 487, 702]}]}
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[{"type": "text", "bbox": [125, 127, 486, 186], "content": "The -factorization corresponding to the nontrivial 4-part of is in case A) and in case B). Note that, by our results from [1], some of these fields have cyclic ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work.", "index": 0}, {"type": "text", "bbox": [125, 187, 487, 211], "content": "The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following", "index": 1}, {"type": "text", "bbox": [125, 217, 487, 290], "content": "Theorem 1. Let be a complex quadratic number field with , and let be its 2-class field. Then rank if and only if disc is the product of three prime discriminants and such that , , and , where is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of such that .", "index": 2}, {"type": "text", "bbox": [125, 296, 486, 344], "content": "This result is the first step in the classification of imaginary quadratic number fields with rank ; it remains to solve these problems for fields with rank and those with since we know that rank whenever rank (using Schur multipliers as in [1]).", "index": 3}, {"type": "text", "bbox": [124, 344, 486, 380], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields , arranged by decreasing value of their discriminants, with ran and noncyclic .", "index": 4}, {"type": "table", "bbox": [126, 429, 489, 588], "content": "", "index": 5}, {"type": "text", "bbox": [125, 603, 486, 651], "content": "Here denotes a generating polynomial for a field as in Theorem 1, denotes the rank of . The cases where follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups with satisfy ), see [3].", "index": 6}, {"type": "text", "bbox": [124, 651, 486, 699], "content": "In order to verify that has rank at least 3 for it is sufficient to show that its genus class field has class group : in fact, then contains a quotient of by , and the claim follows.", "index": 7}]
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[{"bbox": [126, 128, 484, 141], "content": "The -factorization corresponding to the nontrivial 4-part of is", "parent_index": 0, "line_index": 0}, {"bbox": [126, 141, 486, 153], "content": "in case A) and in case B). Note that, by our results from [1], some", "parent_index": 0, "line_index": 1}, {"bbox": [126, 153, 486, 165], "content": "of these fields have cyclic ; however, we do not exclude them right from the", "parent_index": 0, "line_index": 2}, {"bbox": [126, 166, 486, 177], "content": "start since there is no extra work involved and since it provides a welcome check", "parent_index": 0, "line_index": 3}, {"bbox": [126, 178, 213, 188], "content": "on our earlier work.", "parent_index": 0, "line_index": 4}, {"bbox": [137, 188, 488, 201], "content": "The main result of the paper is that rank only occurs for fields of", "parent_index": 1, "line_index": 0}, {"bbox": [125, 200, 333, 214], "content": "type B); more precisely, we prove the following", "parent_index": 1, "line_index": 1}, {"bbox": [126, 219, 487, 232], "content": "Theorem 1. Let be a complex quadratic number field with , and", "parent_index": 2, "line_index": 0}, {"bbox": [125, 231, 487, 244], "content": "let be its 2-class field. Then rank if and only if disc is", "parent_index": 2, "line_index": 1}, {"bbox": [126, 244, 487, 255], "content": "the product of three prime discriminants and such that", "parent_index": 2, "line_index": 2}, {"bbox": [126, 256, 487, 268], "content": ", , and , where is a nonnormal", "parent_index": 2, "line_index": 3}, {"bbox": [126, 268, 487, 281], "content": "quartic subfield of one of the two unramified cyclic quartic extensions of such that", "parent_index": 2, "line_index": 4}, {"bbox": [126, 280, 198, 291], "content": ".", "parent_index": 2, "line_index": 5}, {"bbox": [137, 298, 486, 311], "content": "This result is the first step in the classification of imaginary quadratic number", "parent_index": 3, "line_index": 0}, {"bbox": [126, 310, 486, 322], "content": "fields with rank ; it remains to solve these problems for fields with", "parent_index": 3, "line_index": 1}, {"bbox": [126, 322, 485, 335], "content": "rank and those with since we know that rank", "parent_index": 3, "line_index": 2}, {"bbox": [127, 335, 388, 346], "content": "whenever rank (using Schur multipliers as in [1]).", "parent_index": 3, "line_index": 3}, {"bbox": [137, 346, 486, 359], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "parent_index": 4, "line_index": 0}, {"bbox": [125, 357, 487, 372], "content": "of the first 12 imaginary quadratic fields , arranged by decreasing value of their", "parent_index": 4, "line_index": 1}, {"bbox": [126, 370, 382, 382], "content": "discriminants, with ran and noncyclic .", "parent_index": 4, "line_index": 2}, {"bbox": [137, 605, 486, 618], "content": "Here denotes a generating polynomial for a field as in Theorem 1, denotes", "parent_index": 6, "line_index": 0}, {"bbox": [126, 618, 484, 629], "content": "the rank of . The cases where follow from our theorem combined with", "parent_index": 6, "line_index": 1}, {"bbox": [126, 629, 485, 642], "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "parent_index": 6, "line_index": 2}, {"bbox": [126, 641, 431, 653], "content": "that finite 2-groups with satisfy ), see [3].", "parent_index": 6, "line_index": 3}, {"bbox": [136, 652, 486, 665], "content": "In order to verify that has rank at least 3 for it is", "parent_index": 7, "line_index": 0}, {"bbox": [126, 664, 486, 679], "content": "sufficient to show that its genus class field has class group : in fact,", "parent_index": 7, "line_index": 1}, {"bbox": [126, 676, 487, 690], "content": "then contains a quotient of by , and the claim", "parent_index": 7, "line_index": 2}, {"bbox": [125, 689, 159, 702], "content": "follows.", "parent_index": 7, "line_index": 3}]
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[]
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[{"bbox": [146, 131, 157, 140], "content": "C_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 130, 421, 141], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [433, 131, 484, 140], "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [195, 143, 252, 151], "content": "d=d_{1}d_{2}\\cdot d_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [238, 154, 270, 164], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 0, "subtype": "inline"}, {"bbox": [325, 189, 377, 200], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [205, 221, 211, 229], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [392, 221, 463, 231], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [140, 232, 150, 241], "content": "k^{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [289, 232, 340, 243], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [423, 233, 474, 242], "content": "k=d_{1}d_{2}d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [310, 245, 356, 254], "content": "d_{1},d_{2}\\,>\\,0", "parent_index": 2, "subtype": "inline"}, {"bbox": [380, 245, 440, 254], "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 257, 231, 267], "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [238, 257, 297, 267], "content": "(d_{1}/p_{2})=-1", "parent_index": 2, "subtype": "inline"}, {"bbox": [323, 257, 370, 267], "content": "h_{2}(K)=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 257, 414, 264], "content": "K", "parent_index": 2, "subtype": "inline"}, {"bbox": [437, 269, 443, 277], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 280, 194, 291], "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 312, 158, 319], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [207, 311, 259, 322], "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [147, 324, 193, 334], "content": "\\mathrm{Cl}_{2}(k)=3", "parent_index": 3, "subtype": "inline"}, {"bbox": [262, 324, 326, 334], "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "parent_index": 3, "subtype": "inline"}, {"bbox": [434, 323, 485, 334], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "parent_index": 3, "subtype": "inline"}, {"bbox": [191, 335, 237, 346], "content": "\\mathrm{Cl}_{2}(k)\\geq4", "parent_index": 3, "subtype": "inline"}, {"bbox": [309, 360, 314, 367], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [229, 371, 280, 381], "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [346, 371, 378, 381], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [161, 607, 167, 617], "content": "f", "parent_index": 6, "subtype": "inline"}, {"bbox": [357, 607, 366, 614], "content": "K", "parent_index": 6, "subtype": "inline"}, {"bbox": [444, 610, 449, 614], "content": "r", "parent_index": 6, "subtype": "inline"}, {"bbox": [176, 618, 208, 629], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 6, "subtype": "inline"}, {"bbox": [288, 619, 311, 626], "content": "r=3", "parent_index": 6, "subtype": "inline"}, {"bbox": [213, 643, 221, 650], "content": "G", "parent_index": 6, "subtype": "inline"}, {"bbox": [247, 642, 306, 653], "content": "G/G^{\\prime}\\simeq(2,4)", "parent_index": 6, "subtype": "inline"}, {"bbox": [340, 642, 391, 651], "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "parent_index": 6, "subtype": "inline"}, {"bbox": [243, 654, 276, 665], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [387, 653, 462, 665], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "parent_index": 7, "subtype": "inline"}, {"bbox": [319, 667, 337, 677], "content": "k_{\\mathrm{gen}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [413, 666, 444, 677], "content": "(4,4,8)", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 677, 158, 689], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [280, 678, 312, 689], "content": "(4,4,8)", "parent_index": 7, "subtype": "inline"}, {"bbox": [329, 677, 420, 689], "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "parent_index": 7, "subtype": "inline"}]
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[{"bbox": [285, 392, 325, 403], "content": "", "parent_index": 5, "subtype": "caption"}, {"bbox": [126, 429, 489, 588], "content": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "parent_index": 5, "subtype": "body"}]
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We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\mathrm{Cl}_{2}(k^{1})\geq3$ . Of course, it would be interesting to determine the length of their towers.
The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1.
# 2. Group Theoretic Preliminaries
Let $G$ be a group. If $x,y\ \in\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\{[a,b]:a\in A,b\in B\}$ . The lower central series $\{G_{j}\}$ of $G$ is defined inductively by: $G_{1}\,=\,G$ and $G_{j+1}\,=\,[G,G_{j}]$ for $j~\geq~1$ . The derived series $\{G^{(n)}\}$ is defined inductively by: $G^{(0)}\ =\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\prime}$ , of $G$ .
Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\mathrm{ab}}=G/G^{\prime}$ is of type $(2,2^{m})$ for some positive integer ${\boldsymbol{r}}n$ (necessarily $\geq2$ ). Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{2}$ (actually $\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\geq2$ .
Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\prime})\,=\,n$ where $d(G^{\prime})$ denotes the minimal number of generators of the derived group $G^{\prime}=G_{2}$ of $G$ . Then
$$
G^{\prime}=\langle c_{2},c_{3},\cdot\cdot\cdot,c_{n+1}\rangle;
$$
moreover,
$$
G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle.
$$
Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\Phi(G))$ , where $\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$ . 口
Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and denote the index $\left(G^{\prime}:H^{\prime}\right)$ by $2^{\kappa}$ . Then $G^{\prime}$ contains an element of order $2^{\kappa}$ .
Proof. Without loss of generality, let $H\,=\,\left\langle{b,G^{\prime}}\right\rangle$ . Notice that $G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$ and by our presentation of $H$ , $H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$ . Thus, $G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$ . But since $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ , the order of $c_{2}$ is $\geq2^{\kappa}$ . This establishes the lemma. 口
Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and assume that $(G^{\prime}\,:\,H^{\prime})\;\equiv\;$ 0 mod 4. If $d(G^{\prime})=2$ , then $G_{2}=\langle c_{2},c_{3}\rangle$ and $G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$ for $j>2$ .
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<html><body>
<p data-bbox="124 111 486 195">We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\mathrm{Cl}_{2}(k^{1})\geq3$ . Of course, it would be interesting to determine the length of their towers. </p>
<p data-bbox="124 196 486 255">The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. </p>
<h1 data-bbox="218 263 393 276">2. Group Theoretic Preliminaries </h1>
<p data-bbox="124 281 486 367">Let $G$ be a group. If $x,y\ \in\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\{[a,b]:a\in A,b\in B\}$ . The lower central series $\{G_{j}\}$ of $G$ is defined inductively by: $G_{1}\,=\,G$ and $G_{j+1}\,=\,[G,G_{j}]$ for $j~\geq~1$ . The derived series $\{G^{(n)}\}$ is defined inductively by: $G^{(0)}\ =\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\prime}$ , of $G$ . </p>
<p data-bbox="125 368 486 417">Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\mathrm{ab}}=G/G^{\prime}$ is of type $(2,2^{m})$ for some positive integer ${\boldsymbol{r}}n$ (necessarily $\geq2$ ). Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{2}$ (actually $\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\geq2$ . </p>
<p data-bbox="124 421 486 457">Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\prime})\,=\,n$ where $d(G^{\prime})$ denotes the minimal number of generators of the derived group $G^{\prime}=G_{2}$ of $G$ . Then </p>
<div class="equation" data-bbox="255 464 355 475">$$
G^{\prime}=\langle c_{2},c_{3},\cdot\cdot\cdot,c_{n+1}\rangle;
$$</div>
<p data-bbox="126 479 169 489">moreover, </p>
<div class="equation" data-bbox="230 495 381 507">$$
G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle.
$$</div>
<p data-bbox="125 511 486 573">Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\Phi(G))$ , where $\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$ . 口 </p>
<p data-bbox="125 577 487 614">Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and denote the index $\left(G^{\prime}:H^{\prime}\right)$ by $2^{\kappa}$ . Then $G^{\prime}$ contains an element of order $2^{\kappa}$ . </p>
<p data-bbox="125 619 487 657">Proof. Without loss of generality, let $H\,=\,\left\langle{b,G^{\prime}}\right\rangle$ . Notice that $G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$ and by our presentation of $H$ , $H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$ . Thus, $G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$ . But since $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ , the order of $c_{2}$ is $\geq2^{\kappa}$ . This establishes the lemma. 口 </p>
<p data-bbox="125 661 486 701">Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and assume that $(G^{\prime}\,:\,H^{\prime})\;\equiv\;$ 0 mod 4. If $d(G^{\prime})=2$ , then $G_{2}=\langle c_{2},c_{3}\rangle$ and $G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$ for $j>2$ . </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ . Of course, it would be interesting to determine the length of their towers. ", "page_idx": 2}, {"type": "text", "text": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. ", "page_idx": 2}, {"type": "text", "text": "2. Group Theoretic Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let $G$ be a group. If $x,y\\ \\in\\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\\{[a,b]:a\\in A,b\\in B\\}$ . The lower central series $\\{G_{j}\\}$ of $G$ is defined inductively by: $G_{1}\\,=\\,G$ and $G_{j+1}\\,=\\,[G,G_{j}]$ for $j~\\geq~1$ . The derived series $\\{G^{(n)}\\}$ is defined inductively by: $G^{(0)}\\ =\\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\\prime}$ , of $G$ . ", "page_idx": 2}, {"type": "text", "text": "Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\\mathrm{ab}}=G/G^{\\prime}$ is of type $(2,2^{m})$ for some positive integer ${\\boldsymbol{r}}n$ (necessarily $\\geq2$ ). Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{2}$ (actually $\\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\\geq2$ . ", "page_idx": 2}, {"type": "text", "text": "Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\\prime})\\,=\\,n$ where $d(G^{\\prime})$ denotes the minimal number of generators of the derived group $G^{\\prime}=G_{2}$ of $G$ . Then ", "page_idx": 2}, {"type": "equation", "text": "$$\nG^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "moreover, ", "page_idx": 2}, {"type": "equation", "text": "$$\nG_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\\Phi(G))$ , where $\\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle$ . 口 ", "page_idx": 2}, {"type": "text", "text": "Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and denote the index $\\left(G^{\\prime}:H^{\\prime}\\right)$ by $2^{\\kappa}$ . Then $G^{\\prime}$ contains an element of order $2^{\\kappa}$ . ", "page_idx": 2}, {"type": "text", "text": "Proof. Without loss of generality, let $H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle$ . Notice that $G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle$ and by our presentation of $H$ , $H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle$ . Thus, $G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle$ . But since $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ , the order of $c_{2}$ is $\\geq2^{\\kappa}$ . This establishes the lemma. 口 ", "page_idx": 2}, {"type": "text", "text": "Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and assume that $(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;$ 0 mod 4. If $d(G^{\\prime})=2$ , then $G_{2}=\\langle c_{2},c_{3}\\rangle$ and $G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle$ for $j>2$ . ", "page_idx": 2}]
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It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "口", "type": "text"}], "index": 34}], "index": 32}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "口", "type": "text"}], "index": 40}], "index": 39}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 98], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 486, 195], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 486, 126], "score": 1.0, "content": "We mention one last feature gleaned from the table. It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3, "bbox_fs": [125, 114, 486, 198]}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9, "bbox_fs": [125, 197, 488, 257]}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16, "bbox_fs": [124, 284, 487, 370]}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5, "bbox_fs": [124, 368, 486, 419]}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25, "bbox_fs": [125, 423, 487, 460]}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [126, 481, 169, 491]}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "口", "type": "text"}], "index": 34}], "index": 32, "bbox_fs": [124, 514, 486, 575]}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [126, 580, 486, 615]}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "口", "type": "text"}], "index": 40}], "index": 39, "bbox_fs": [126, 621, 487, 658]}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42, "bbox_fs": [124, 664, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 486, 195], "content": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers.", "index": 0}, {"type": "text", "bbox": [124, 196, 486, 255], "content": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1.", "index": 1}, {"type": "title", "bbox": [218, 263, 393, 276], "content": "2. Group Theoretic Preliminaries", "index": 2}, {"type": "text", "bbox": [124, 281, 486, 367], "content": "Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of .", "index": 3}, {"type": "text", "bbox": [125, 368, 486, 417], "content": "Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for .", "index": 4}, {"type": "text", "bbox": [124, 421, 486, 457], "content": "Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then", "index": 5}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "content": "", "index": 6}, {"type": "text", "bbox": [126, 479, 169, 489], "content": "moreover,", "index": 7}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "content": "", "index": 8}, {"type": "text", "bbox": [125, 511, 486, 573], "content": "Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口", "index": 9}, {"type": "text", "bbox": [125, 577, 487, 614], "content": "Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order .", "index": 10}, {"type": "text", "bbox": [125, 619, 487, 657], "content": "Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口", "index": 11}, {"type": "text", "bbox": [125, 661, 486, 701], "content": "Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. If , then and for .", "index": 12}]
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[{"bbox": [137, 114, 486, 126], "content": "We mention one last feature gleaned from the table. It follows from conditional", "parent_index": 0, "line_index": 0}, {"bbox": [125, 126, 486, 139], "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 150], "content": "dratic fields with rank and discriminant have finite class", "parent_index": 0, "line_index": 2}, {"bbox": [125, 150, 486, 162], "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "parent_index": 0, "line_index": 3}, {"bbox": [126, 163, 485, 174], "content": "that those with discriminants , and have finite (2-)class field", "parent_index": 0, "line_index": 4}, {"bbox": [125, 174, 485, 186], "content": "tower even though rank . Of course, it would be interesting to determine", "parent_index": 0, "line_index": 5}, {"bbox": [126, 186, 240, 198], "content": "the length of their towers.", "parent_index": 0, "line_index": 6}, {"bbox": [137, 197, 485, 210], "content": "The structure of this paper is as follows: we use results from group theory", "parent_index": 1, "line_index": 0}, {"bbox": [125, 209, 487, 222], "content": "developed in Section 2 to pull down the condition rank from the field", "parent_index": 1, "line_index": 1}, {"bbox": [126, 220, 488, 235], "content": "with degree to a subfield of with degree 8. Using the arithmetic of", "parent_index": 1, "line_index": 2}, {"bbox": [125, 233, 486, 247], "content": "dihedral fields from Section 4 we then go down to the field of degree 4 occurring", "parent_index": 1, "line_index": 3}, {"bbox": [126, 246, 188, 257], "content": "in Theorem 1.", "parent_index": 1, "line_index": 4}, {"bbox": [217, 266, 394, 277], "content": "2. Group Theoretic Preliminaries", "parent_index": 2, "line_index": 0}, {"bbox": [136, 284, 486, 296], "content": "Let be a group. If , then we let denote the", "parent_index": 3, "line_index": 0}, {"bbox": [125, 296, 484, 309], "content": "commutator of and . If and are nonempty subsets of , then", "parent_index": 3, "line_index": 1}, {"bbox": [126, 308, 486, 320], "content": "denotes the subgroup of generated by the set . The lower", "parent_index": 3, "line_index": 2}, {"bbox": [124, 320, 485, 334], "content": "central series of is defined inductively by: and", "parent_index": 3, "line_index": 3}, {"bbox": [124, 332, 487, 345], "content": "for . The derived series is defined inductively by: and", "parent_index": 3, "line_index": 4}, {"bbox": [126, 342, 487, 359], "content": "for . Notice that the commutator", "parent_index": 3, "line_index": 5}, {"bbox": [125, 357, 212, 370], "content": "subgroup, , of .", "parent_index": 3, "line_index": 6}, {"bbox": [137, 368, 486, 383], "content": "Throughout this section, we assume that is a finite, nonmetacyclic, 2-group", "parent_index": 4, "line_index": 0}, {"bbox": [125, 379, 486, 394], "content": "such that its abelianization is of type for some positive integer", "parent_index": 4, "line_index": 1}, {"bbox": [126, 392, 484, 406], "content": "(necessarily ). Let , where mod (actually", "parent_index": 4, "line_index": 2}, {"bbox": [124, 404, 440, 419], "content": "since is nonmetacyclic, cf. [1]); and for .", "parent_index": 4, "line_index": 3}, {"bbox": [125, 423, 487, 435], "content": "Lemma 1. Let be as above (but not necessarily metabelian). Suppose that", "parent_index": 5, "line_index": 0}, {"bbox": [126, 434, 487, 448], "content": "where denotes the minimal number of generators of the derived", "parent_index": 5, "line_index": 1}, {"bbox": [126, 447, 245, 460], "content": "group of . Then", "parent_index": 5, "line_index": 2}, {"bbox": [126, 481, 169, 491], "content": "moreover,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 514, 486, 526], "content": "Proof. By the Burnside Basis Theorem, , where is the", "parent_index": 9, "line_index": 0}, {"bbox": [126, 526, 485, 538], "content": "Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5].", "parent_index": 9, "line_index": 1}, {"bbox": [124, 537, 485, 551], "content": "But in the case of a 2-group, , see [8]. By Blackburn, [3], since", "parent_index": 9, "line_index": 2}, {"bbox": [124, 549, 485, 564], "content": "has elementary derived group, we know that .", "parent_index": 9, "line_index": 3}, {"bbox": [126, 560, 486, 575], "content": "Again, by the Burnside Basis Theorem, . 口", "parent_index": 9, "line_index": 4}, {"bbox": [126, 580, 486, 592], "content": "Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a", "parent_index": 10, "line_index": 0}, {"bbox": [126, 592, 486, 604], "content": "maximal subgroup of such that is cyclic, and denote the index by", "parent_index": 10, "line_index": 1}, {"bbox": [126, 603, 327, 615], "content": ". Then contains an element of order .", "parent_index": 10, "line_index": 2}, {"bbox": [126, 621, 485, 634], "content": "Proof. Without loss of generality, let . Notice that", "parent_index": 11, "line_index": 0}, {"bbox": [126, 633, 487, 646], "content": "and by our presentation of , . Thus, . But since", "parent_index": 11, "line_index": 1}, {"bbox": [126, 646, 486, 658], "content": ", the order of is . This establishes the lemma. 口", "parent_index": 11, "line_index": 2}, {"bbox": [125, 664, 487, 676], "content": "Lemma 3. Let be as above and again assume is metabelian. Let be a", "parent_index": 12, "line_index": 0}, {"bbox": [126, 675, 486, 687], "content": "maximal subgroup of such that is cyclic, and assume that", "parent_index": 12, "line_index": 1}, {"bbox": [124, 687, 454, 702], "content": "0 mod 4. If , then and for .", "parent_index": 12, "line_index": 2}]
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[{"bbox": [222, 139, 273, 150], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [351, 140, 415, 148], "content": "0>d>-2000", "parent_index": 0, "subtype": "inline"}, {"bbox": [173, 164, 179, 171], "content": "k", "parent_index": 0, "subtype": "inline"}, {"bbox": [266, 164, 294, 172], "content": "-1015", "parent_index": 0, "subtype": "inline"}, {"bbox": [298, 164, 326, 172], "content": "-1595", "parent_index": 0, "subtype": "inline"}, {"bbox": [348, 164, 376, 172], "content": "-1780", "parent_index": 0, "subtype": "inline"}, {"bbox": [228, 175, 279, 186], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [370, 210, 422, 221], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 222, 136, 231], "content": "k^{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [194, 223, 217, 231], "content": "2^{m+2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [279, 224, 286, 231], "content": "L", "parent_index": 1, "subtype": "inline"}, {"bbox": [301, 222, 311, 231], "content": "k^{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [381, 236, 391, 243], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [157, 286, 165, 293], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [244, 286, 285, 295], "content": "x,y\\ \\in\\ G", "parent_index": 3, "subtype": "inline"}, {"bbox": [349, 285, 433, 296], "content": "[x,y]~=~x^{-1}y^{-1}x y", "parent_index": 3, "subtype": "inline"}, {"bbox": [196, 300, 203, 305], "content": "x", "parent_index": 3, "subtype": "inline"}, {"bbox": [228, 300, 234, 307], "content": "_y", "parent_index": 3, "subtype": "inline"}, {"bbox": [257, 298, 265, 305], "content": "A", "parent_index": 3, "subtype": "inline"}, {"bbox": [290, 298, 299, 305], "content": "B", "parent_index": 3, "subtype": "inline"}, {"bbox": [419, 298, 427, 305], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [459, 297, 484, 308], "content": "[A,B]", "parent_index": 3, "subtype": "inline"}, {"bbox": [235, 310, 243, 317], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [339, 309, 433, 320], "content": "\\{[a,b]:a\\in A,b\\in B\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [187, 321, 209, 332], "content": "\\{G_{j}\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [226, 322, 234, 329], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [356, 322, 392, 331], "content": "G_{1}\\,=\\,G", "parent_index": 3, "subtype": "inline"}, {"bbox": [416, 321, 485, 332], "content": "G_{j+1}\\,=\\,[G,G_{j}]", "parent_index": 3, "subtype": "inline"}, {"bbox": [142, 335, 170, 344], "content": "j~\\geq~1", "parent_index": 3, "subtype": "inline"}, {"bbox": [267, 333, 297, 344], "content": "\\{G^{(n)}\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [420, 333, 464, 342], "content": "G^{(0)}\\ =\\ G", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 345, 219, 357], "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "parent_index": 3, "subtype": "inline"}, {"bbox": [238, 347, 264, 356], "content": "n\\geq0", "parent_index": 3, "subtype": "inline"}, {"bbox": [325, 345, 410, 357], "content": "G^{(1)}=G_{2}=[G,G]", "parent_index": 3, "subtype": "inline"}, {"bbox": [172, 358, 183, 366], "content": "G^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [200, 359, 208, 366], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [323, 371, 331, 378], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [246, 382, 300, 393], "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [346, 382, 376, 393], "content": "(2,2^{m})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 398, 135, 402], "content": "{\\boldsymbol{r}}n", "parent_index": 4, "subtype": "inline"}, {"bbox": [191, 396, 207, 403], "content": "\\geq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [236, 394, 279, 405], "content": "G=\\langle a,b\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [313, 393, 370, 402], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "parent_index": 4, "subtype": "inline"}, {"bbox": [394, 395, 407, 404], "content": "G_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [452, 395, 484, 404], "content": "\\mathrm{mod}G_{3}", "parent_index": 4, "subtype": "inline"}, {"bbox": [150, 407, 158, 414], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [275, 406, 316, 417], "content": "c_{2}=[a,b]", "parent_index": 4, "subtype": "inline"}, {"bbox": [339, 406, 394, 417], "content": "c_{j+1}=[b,c_{j}]", "parent_index": 4, "subtype": "inline"}, {"bbox": [412, 407, 435, 416], "content": "j\\geq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [200, 425, 208, 432], "content": "G", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 436, 172, 447], "content": "d(G^{\\prime})\\,=\\,n", "parent_index": 5, "subtype": "inline"}, {"bbox": [205, 436, 228, 447], "content": "d(G^{\\prime})", "parent_index": 5, "subtype": "inline"}, {"bbox": [153, 449, 190, 458], "content": "G^{\\prime}=G_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [205, 449, 213, 456], "content": "G", "parent_index": 5, "subtype": "inline"}, {"bbox": [255, 464, 355, 475], "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "parent_index": 6, "subtype": "interline"}, {"bbox": [230, 495, 381, 507], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "parent_index": 8, "subtype": "interline"}, {"bbox": [305, 515, 398, 525], "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "parent_index": 9, "subtype": "inline"}, {"bbox": [434, 515, 457, 525], "content": "\\Phi(G)", "parent_index": 9, "subtype": "inline"}, {"bbox": [216, 528, 224, 535], "content": "G", "parent_index": 9, "subtype": "inline"}, {"bbox": [442, 528, 450, 535], "content": "G", "parent_index": 9, "subtype": "inline"}, {"bbox": [258, 538, 308, 549], "content": "\\Phi(G)=G^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [460, 538, 485, 549], "content": "G/G_{2}^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [329, 550, 482, 561], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [302, 563, 388, 573], "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [199, 582, 207, 589], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [355, 582, 363, 589], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [453, 582, 462, 589], "content": "H", "parent_index": 10, "subtype": "inline"}, {"bbox": [218, 594, 226, 601], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [272, 593, 295, 603], "content": "H/G^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [434, 592, 472, 603], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 605, 136, 613], "content": "2^{\\kappa}", "parent_index": 10, "subtype": "inline"}, {"bbox": [170, 605, 180, 613], "content": "G^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [312, 605, 323, 613], "content": "2^{\\kappa}", "parent_index": 10, "subtype": "inline"}, {"bbox": [294, 622, 347, 633], "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [412, 621, 485, 633], "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 636, 254, 643], "content": "H", "parent_index": 11, "subtype": "inline"}, {"bbox": [260, 635, 332, 645], "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [367, 634, 437, 645], "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [126, 647, 188, 657], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "parent_index": 11, "subtype": "inline"}, {"bbox": [249, 650, 258, 656], "content": "c_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [271, 648, 291, 656], "content": "\\geq2^{\\kappa}", "parent_index": 11, "subtype": "inline"}, {"bbox": [199, 666, 207, 673], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [352, 664, 361, 673], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [452, 664, 462, 673], "content": "H", "parent_index": 12, "subtype": "inline"}, {"bbox": [222, 677, 230, 685], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [279, 676, 304, 687], "content": "H/G^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [430, 675, 486, 687], "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "parent_index": 12, "subtype": "inline"}, {"bbox": [179, 690, 221, 701], "content": "d(G^{\\prime})=2", "parent_index": 12, "subtype": "inline"}, {"bbox": [249, 690, 305, 701], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "parent_index": 12, "subtype": "inline"}, {"bbox": [327, 687, 408, 701], "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "parent_index": 12, "subtype": "inline"}, {"bbox": [426, 689, 451, 700], "content": "j>2", "parent_index": 12, "subtype": "inline"}]
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Proof. Assume that $d(G^{\prime})=2$ . By Lemma 1, $G_{2}=\langle c_{2},c_{3}\rangle$ and hence $c_{4}\in\langle c_{2},c_{3}\rangle$ . Write $c_{4}\:=\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\langle b,c_{2},c_{3}\rangle$ and write $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ for some $\kappa\geq2$ . Since $c_{3},c_{4}\in H^{\prime}$ we have, $c_{2}^{x}\equiv1$ mod $H^{\prime}$ . By the proof of Lemma 2, this implies that $x\equiv0$ mod $2^{\kappa}$ . Write $x\,=\,2^{\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$ we see that $c_{3}^{y}\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\in G_{4}$ . This, however, implies that $G_{2}=\langle c_{2}\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$ , cf. [1], we obtain the lemma. 口
Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\kappa}$ and $\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\kappa}$ , where $2^{\kappa}$ is the class number of the quadratic subfield $\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\kappa\geq2$ , and the assumption $\left(G^{\prime}:H^{\prime}\right)\geq4$ is always satisfied for the fields that we consider.
Table 2
<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>
We now use the above results to prove the following useful proposition.
Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\prime}\;\simeq\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\prime}$ and $K/G^{\prime}$ are cyclic. Moreover, assume that $(G^{\prime}:H^{\prime})\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then
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<html><body>
<p data-bbox="125 111 487 237">Proof. Assume that $d(G^{\prime})=2$ . By Lemma 1, $G_{2}=\langle c_{2},c_{3}\rangle$ and hence $c_{4}\in\langle c_{2},c_{3}\rangle$ . Write $c_{4}\:=\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\langle b,c_{2},c_{3}\rangle$ and write $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ for some $\kappa\geq2$ . Since $c_{3},c_{4}\in H^{\prime}$ we have, $c_{2}^{x}\equiv1$ mod $H^{\prime}$ . By the proof of Lemma 2, this implies that $x\equiv0$ mod $2^{\kappa}$ . Write $x\,=\,2^{\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$ we see that $c_{3}^{y}\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\in G_{4}$ . This, however, implies that $G_{2}=\langle c_{2}\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$ , cf. [1], we obtain the lemma. 口 </p>
<p data-bbox="125 246 487 366">Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\kappa}$ and $\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\kappa}$ , where $2^{\kappa}$ is the class number of the quadratic subfield $\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\kappa\geq2$ , and the assumption $\left(G^{\prime}:H^{\prime}\right)\geq4$ is always satisfied for the fields that we consider. </p>
<div class="table" data-bbox="167 416 444 622"><p class="caption" data-bbox="285 377 327 389">Table 2 </p><table data-bbox="167 416 444 622"><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></div>
<p data-bbox="137 643 450 656">We now use the above results to prove the following useful proposition. </p>
<p data-bbox="125 662 486 700">Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\prime}\;\simeq\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\prime}$ and $K/G^{\prime}$ are cyclic. Moreover, assume that $(G^{\prime}:H^{\prime})\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then </p>
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0003244v1
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[{"type": "text", "text": "Proof. Assume that $d(G^{\\prime})=2$ . By Lemma 1, $G_{2}=\\langle c_{2},c_{3}\\rangle$ and hence $c_{4}\\in\\langle c_{2},c_{3}\\rangle$ . Write $c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\\langle b,c_{2},c_{3}\\rangle$ and write $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ for some $\\kappa\\geq2$ . Since $c_{3},c_{4}\\in H^{\\prime}$ we have, $c_{2}^{x}\\equiv1$ mod $H^{\\prime}$ . By the proof of Lemma 2, this implies that $x\\equiv0$ mod $2^{\\kappa}$ . Write $x\\,=\\,2^{\\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}$ we see that $c_{3}^{y}\\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\\in G_{4}$ . This, however, implies that $G_{2}=\\langle c_{2}\\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle$ , cf. [1], we obtain the lemma. 口 ", "page_idx": 3}, {"type": "text", "text": "Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\\kappa}$ and $\\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\\kappa}$ , where $2^{\\kappa}$ is the class number of the quadratic subfield $\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\\kappa\\geq2$ , and the assumption $\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4$ is always satisfied for the fields that we consider. ", "page_idx": 3}, {"type": "table", "img_path": "images/20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg", "table_caption": ["Table 2 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "We now use the above results to prove the following useful proposition. ", "page_idx": 3}, {"type": "text", "text": "Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\\prime}\\;\\simeq\\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\\prime}$ and $K/G^{\\prime}$ are cyclic. Moreover, assume that $(G^{\\prime}:H^{\\prime})\\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then ", "page_idx": 3}]
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29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [382, 1788, 1252, 1788, 1252, 1823, 382, 1823], "score": 0.9}, {"category_id": 6, "poly": [793, 1049, 909, 1049, 909, 1083, 793, 1083], "score": 0.808}, {"category_id": 2, "poly": [348, 254, 366, 254, 366, 275, 348, 275], "score": 0.753}, {"category_id": 13, "poly": [431, 355, 552, 355, 552, 384, 431, 384], "score": 0.95, "latex": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}"}, {"category_id": 13, "poly": [907, 322, 1061, 322, 1061, 351, 907, 351], "score": 0.95, "latex": "G_{2}=\\langle 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1248.0, 1793.0, 1248.0, 1827.0, 385.0, 1827.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [793.0, 1052.0, 909.0, 1052.0, 909.0, 1088.0, 793.0, 1088.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [349.0, 263.5, 367.0, 263.5, 367.0, 276.5, 349.0, 276.5], "score": 1.0, "text": ""}], "page_info": {"page_no": 3, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22κx1c23y1. Consequently, by induction we have cj ∈", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j−", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "口", "type": "text"}], "index": 9}], "index": 4.5}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}], "index": 39}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [125, 94, 132, 99], "spans": [{"bbox": [125, 94, 132, 99], "score": 1.0, "content": "4", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22κx1c23y1. Consequently, by induction we have cj ∈", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j−", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "口", "type": "text"}], "index": 9}], "index": 4.5, "bbox_fs": [123, 114, 489, 238]}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5, "bbox_fs": [124, 248, 487, 368]}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37, "bbox_fs": [138, 645, 449, 657]}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}, {"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text", "cross_page": true}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text", "cross_page": true}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text", "cross_page": true}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10, "cross_page": true}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text", "cross_page": true}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text", "cross_page": true}], "index": 0}], "index": 39, "bbox_fs": [125, 665, 487, 701]}]}
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[{"type": "text", "bbox": [125, 111, 487, 237], "content": "Proof. Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口", "index": 0}, {"type": "text", "bbox": [125, 246, 487, 366], "content": "Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider.", "index": 1}, {"type": "table", "bbox": [167, 416, 444, 622], "content": "", "index": 2}, {"type": "text", "bbox": [137, 643, 450, 656], "content": "We now use the above results to prove the following useful proposition.", "index": 3}, {"type": "text", "bbox": [125, 662, 486, 700], "content": "Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then", "index": 4}]
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[{"bbox": [126, 114, 485, 126], "content": "Proof. Assume that . By Lemma 1, and hence .", "parent_index": 0, "line_index": 0}, {"bbox": [123, 122, 489, 143], "content": "Write where are positive integers. Without loss of generality, let", "parent_index": 0, "line_index": 1}, {"bbox": [126, 136, 487, 152], "content": "and write for some . Since we have,", "parent_index": 0, "line_index": 2}, {"bbox": [126, 150, 487, 162], "content": "mod . By the proof of Lemma 2, this implies that mod . Write", "parent_index": 0, "line_index": 3}, {"bbox": [126, 160, 482, 178], "content": "for some positive integer . On the other hand, since", "parent_index": 0, "line_index": 4}, {"bbox": [125, 175, 486, 187], "content": "we see that mod . If were odd, then . This, however, implies", "parent_index": 0, "line_index": 5}, {"bbox": [124, 186, 486, 200], "content": "that , contrary to our assumptions. Thus is even, say . From", "parent_index": 0, "line_index": 6}, {"bbox": [123, 196, 487, 215], "content": "all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈", "parent_index": 0, "line_index": 7}, {"bbox": [123, 208, 485, 232], "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1],", "parent_index": 0, "line_index": 8}, {"bbox": [125, 226, 486, 238], "content": "we obtain the lemma. 口", "parent_index": 0, "line_index": 9}, {"bbox": [137, 248, 485, 261], "content": "Let us translate the above into the field-theoretic language. Let be an imagi-", "parent_index": 1, "line_index": 0}, {"bbox": [124, 261, 485, 272], "content": "nary quadratic number field of type A) or B) (see the Introduction), and let", "parent_index": 1, "line_index": 1}, {"bbox": [125, 273, 487, 284], "content": "be one of the two quadratic subextensions of over which is cyclic. If", "parent_index": 1, "line_index": 2}, {"bbox": [126, 282, 487, 298], "content": "and , then Lemma 2 implies that contains", "parent_index": 1, "line_index": 3}, {"bbox": [126, 296, 486, 308], "content": "an element of order . Table 2 contains the relevant information for the fields", "parent_index": 1, "line_index": 4}, {"bbox": [125, 307, 486, 321], "content": "occurring in Table 1. An application of the class number formula to (see e.g.", "parent_index": 1, "line_index": 5}, {"bbox": [124, 320, 487, 333], "content": "Proposition 3 below) shows immediately that , where is the class", "parent_index": 1, "line_index": 6}, {"bbox": [124, 332, 486, 346], "content": "number of the quadratic subfield of , where ; in particu-", "parent_index": 1, "line_index": 7}, {"bbox": [125, 344, 486, 357], "content": "lar, we always have , and the assumption is always satisfied for", "parent_index": 1, "line_index": 8}, {"bbox": [125, 356, 244, 368], "content": "the fields that we consider.", "parent_index": 1, "line_index": 9}, {"bbox": [138, 645, 449, 657], "content": "We now use the above results to prove the following useful proposition.", "parent_index": 3, "line_index": 0}, {"bbox": [125, 665, 486, 677], "content": "Proposition 1. Let be a nonmetacyclic 2-group such that ;", "parent_index": 4, "line_index": 0}, {"bbox": [127, 677, 484, 689], "content": "(hence ). Let and be the two maximal subgroups of such that", "parent_index": 4, "line_index": 1}, {"bbox": [127, 689, 487, 701], "content": "and are cyclic. Moreover, assume that mod 4. Finally, assume", "parent_index": 4, "line_index": 2}, {"bbox": [127, 115, 426, 126], "content": "that is a subgroup of index 4 in not contained in or Then", "parent_index": 4, "line_index": 3}]
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[]
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[{"bbox": [215, 115, 258, 126], "content": "d(G^{\\prime})=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [326, 115, 381, 126], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [431, 115, 482, 126], "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [155, 127, 198, 138], "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "parent_index": 0, "subtype": "inline"}, {"bbox": [232, 131, 248, 137], "content": "x,y", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 139, 187, 150], "content": "H=\\langle b,c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [235, 139, 297, 150], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "parent_index": 0, "subtype": "inline"}, {"bbox": [340, 140, 365, 149], "content": "\\kappa\\geq2", "parent_index": 0, "subtype": "inline"}, {"bbox": [398, 139, 444, 149], "content": "c_{3},c_{4}\\in H^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 152, 155, 162], "content": "c_{2}^{x}\\equiv1", "parent_index": 0, "subtype": "inline"}, {"bbox": [179, 151, 191, 159], "content": "H^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 152, 417, 159], "content": "x\\equiv0", "parent_index": 0, "subtype": "inline"}, {"bbox": [442, 152, 452, 159], "content": "2^{\\kappa}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 165, 168, 174], "content": "x\\,=\\,2^{\\kappa}x_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [285, 168, 295, 174], "content": "x_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [420, 162, 482, 175], "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [180, 176, 209, 187], "content": "c_{3}^{y}\\equiv1", "parent_index": 0, "subtype": "inline"}, {"bbox": [233, 177, 245, 186], "content": "G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [264, 180, 270, 186], "content": "_y", "parent_index": 0, "subtype": "inline"}, {"bbox": [343, 177, 378, 186], "content": "c_{3}\\in G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [148, 188, 191, 199], "content": "G_{2}=\\langle c_{2}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [356, 191, 362, 198], "content": "_y", "parent_index": 0, "subtype": "inline"}, {"bbox": [419, 189, 453, 198], "content": "y=2y_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [228, 199, 291, 212], "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [272, 212, 449, 227], "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [421, 250, 426, 257], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [464, 262, 485, 272], "content": "M/k", "parent_index": 1, "subtype": "inline"}, {"bbox": [337, 273, 358, 284], 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"subtype": "inline"}, {"bbox": [331, 334, 342, 342], "content": "M", "parent_index": 1, "subtype": "inline"}, {"bbox": [376, 334, 433, 344], "content": "(d_{i}/p_{j})=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [213, 347, 237, 354], "content": "\\kappa\\geq2", "parent_index": 1, "subtype": "inline"}, {"bbox": [332, 345, 389, 356], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "parent_index": 1, "subtype": "inline"}, {"bbox": [221, 667, 229, 674], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [409, 666, 482, 677], "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "parent_index": 4, "subtype": "inline"}, {"bbox": [158, 679, 186, 687], "content": "m>1", "parent_index": 4, "subtype": "inline"}, {"bbox": [215, 679, 224, 686], "content": "H", "parent_index": 4, "subtype": "inline"}, {"bbox": [247, 679, 257, 686], "content": "K", "parent_index": 4, "subtype": "inline"}, {"bbox": [406, 679, 414, 686], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [460, 678, 484, 689], "content": "H/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 690, 169, 701], "content": "K/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [322, 690, 379, 701], "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "parent_index": 4, "subtype": "inline"}, {"bbox": [146, 116, 156, 123], "content": "N", "parent_index": 4, "subtype": "inline"}, {"bbox": [278, 115, 286, 124], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 115, 374, 123], "content": "H", "parent_index": 4, "subtype": "inline"}, {"bbox": [389, 116, 399, 123], "content": "K", "parent_index": 4, "subtype": "inline"}]
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[{"bbox": [285, 377, 327, 389], "content": "", "parent_index": 2, "subtype": "caption"}, {"bbox": [167, 416, 444, 622], "content": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "parent_index": 2, "subtype": "body"}]
|
|
$$
(N:N^{\prime})\;\left\{\begin{array}{l l}{{=}}&{{2^{m}\;\;\;\;\;i f\,d(G^{\prime})=1}}\\ {{=}}&{{2^{m+1}\;\;\;\;i f\,d(G^{\prime})=2}}\\ {{\geq}}&{{2^{m+2}\;\;\;\;i f\,d(G^{\prime})\geq3}}\end{array}\right..
$$
Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{3}$ . Also let $H=\langle b,G^{\prime}\rangle$ and $K=\langle a b,G^{\prime}\rangle$ (without loss of generality). Then $N=\langle a b^{2},G^{\prime}\rangle$ or $N=\langle a,b^{4},G^{\prime}\rangle$ .
Suppose that $N=\left\langle a b^{2},G^{\prime}\right\rangle$ .
First assume $d(G^{\prime})=1$ . Then $G^{\prime}=\langle c_{2}\rangle$ and thus $N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\eta_{4}$ for some $\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$ , and so $\left(G^{\prime}:N^{\prime}\right)=2$ . Since $(N:G^{\prime})=2^{m-1}$ , we get $(N:N^{\prime})=2^{m}$ as desired.
Next, assume that $d(G^{\prime})\;=\;2$ . Then $N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$ and $[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$ where $\eta_{j}~\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$ and so $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$ . But then $N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\prime}\supseteq G_{4}$ . But notice that $N_{3}\subseteq G_{4}$ . Thus $N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$ and so $(G^{\prime}:N^{\prime})=4$ which in turn implies that $(N:N^{\prime})=2^{m+1}$ , as desired.
Finally, assume $d(G^{\prime})\geq3$ . Then $d(G^{\prime}/G_{5})=3$ . Moreover there exists an exact sequence
$$
N/N^{\prime}\longrightarrow(N/G_{5})/(N/G_{5})^{\prime}\longrightarrow1,
$$
and thus $\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\,=\,1$ which we now assume. $N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$ and so, arguing as above, we have $N^{\prime}=$ $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$ , where $\eta_{j}\;\in\;G_{j}$ . But $N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$ $\langle c_{2}^{4}\rangle$ . Therefore, $N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$ . From this we see that $(G^{\prime}\,:\,N^{\prime})\,=\,8$ and thus $(N:N^{\prime})=2^{m+2}$ as desired.
Now suppose that $N\,=\,\langle a,b^{4},G^{\prime}\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ .
This establishes the proposition.
# 3. Number Theoretic Preliminaries
Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\operatorname{Gal}(M/k)\simeq D_{4}$ .
Proof. R´edei and Reichardt [12] proved this for $k=\mathbb{Q}$ ; the general case is analogous.
We shall make extensive use of the class number formula for extensions of type $(2,2)$ :
Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then
$$
h(K)=2^{d-\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},
$$
where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\kappa$ is the $\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\upsilon=0$ except when $K\subseteq k(\sqrt{E_{k}}\,)$ , where $\upsilon=1$ .
Proof. See [10].
|
<html><body>
<div class="equation" data-bbox="219 129 391 169">$$
(N:N^{\prime})\;\left\{\begin{array}{l l}{{=}}&{{2^{m}\;\;\;\;\;i f\,d(G^{\prime})=1}}\\ {{=}}&{{2^{m+1}\;\;\;\;i f\,d(G^{\prime})=2}}\\ {{\geq}}&{{2^{m+2}\;\;\;\;i f\,d(G^{\prime})\geq3}}\end{array}\right..
$$</div>
<p data-bbox="124 171 486 208">Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{3}$ . Also let $H=\langle b,G^{\prime}\rangle$ and $K=\langle a b,G^{\prime}\rangle$ (without loss of generality). Then $N=\langle a b^{2},G^{\prime}\rangle$ or $N=\langle a,b^{4},G^{\prime}\rangle$ . </p>
<p data-bbox="135 209 261 220">Suppose that $N=\left\langle a b^{2},G^{\prime}\right\rangle$ . </p>
<p data-bbox="125 220 487 256">First assume $d(G^{\prime})=1$ . Then $G^{\prime}=\langle c_{2}\rangle$ and thus $N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\eta_{4}$ for some $\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$ , and so $\left(G^{\prime}:N^{\prime}\right)=2$ . Since $(N:G^{\prime})=2^{m-1}$ , we get $(N:N^{\prime})=2^{m}$ as desired. </p>
<p data-bbox="124 256 486 316">Next, assume that $d(G^{\prime})\;=\;2$ . Then $N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$ and $[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$ where $\eta_{j}~\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$ and so $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$ . But then $N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\prime}\supseteq G_{4}$ . But notice that $N_{3}\subseteq G_{4}$ . Thus $N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$ and so $(G^{\prime}:N^{\prime})=4$ which in turn implies that $(N:N^{\prime})=2^{m+1}$ , as desired. </p>
<p data-bbox="125 316 486 340">Finally, assume $d(G^{\prime})\geq3$ . Then $d(G^{\prime}/G_{5})=3$ . Moreover there exists an exact sequence </p>
<div class="equation" data-bbox="230 348 380 359">$$
N/N^{\prime}\longrightarrow(N/G_{5})/(N/G_{5})^{\prime}\longrightarrow1,
$$</div>
<p data-bbox="124 362 487 422">and thus $\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\,=\,1$ which we now assume. $N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$ and so, arguing as above, we have $N^{\prime}=$ $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$ , where $\eta_{j}\;\in\;G_{j}$ . But $N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$ $\langle c_{2}^{4}\rangle$ . Therefore, $N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$ . From this we see that $(G^{\prime}\,:\,N^{\prime})\,=\,8$ and thus $(N:N^{\prime})=2^{m+2}$ as desired. </p>
<p data-bbox="126 423 486 447">Now suppose that $N\,=\,\langle a,b^{4},G^{\prime}\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . </p>
<p data-bbox="137 447 281 459">This establishes the proposition. </p>
<h1 data-bbox="213 471 397 484">3. Number Theoretic Preliminaries </h1>
<p data-bbox="125 489 486 526">Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\operatorname{Gal}(M/k)\simeq D_{4}$ . </p>
<p data-bbox="125 532 486 545">Proof. R´edei and Reichardt [12] proved this for $k=\mathbb{Q}$ ; the general case is analogous. </p>
<p data-bbox="125 564 487 589">We shall make extensive use of the class number formula for extensions of type $(2,2)$ : </p>
<p data-bbox="125 595 486 619">Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then </p>
<div class="equation" data-bbox="203 626 404 639">$$
h(K)=2^{d-\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},
$$</div>
<p data-bbox="125 644 486 681">where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\kappa$ is the $\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\upsilon=0$ except when $K\subseteq k(\sqrt{E_{k}}\,)$ , where $\upsilon=1$ . </p>
<p data-bbox="126 687 192 700">Proof. See [10]. </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 4}, {"type": "equation", "text": "$$\n(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{3}$ . Also let $H=\\langle b,G^{\\prime}\\rangle$ and $K=\\langle a b,G^{\\prime}\\rangle$ (without loss of generality). Then $N=\\langle a b^{2},G^{\\prime}\\rangle$ or $N=\\langle a,b^{4},G^{\\prime}\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "Suppose that $N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "First assume $d(G^{\\prime})=1$ . Then $G^{\\prime}=\\langle c_{2}\\rangle$ and thus $N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\\eta_{4}$ for some $\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle$ , and so $\\left(G^{\\prime}:N^{\\prime}\\right)=2$ . Since $(N:G^{\\prime})=2^{m-1}$ , we get $(N:N^{\\prime})=2^{m}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Next, assume that $d(G^{\\prime})\\;=\\;2$ . Then $N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}$ and $[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}$ where $\\eta_{j}~\\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle$ and so $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}$ . But then $N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\\prime}\\supseteq G_{4}$ . But notice that $N_{3}\\subseteq G_{4}$ . Thus $N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle$ and so $(G^{\\prime}:N^{\\prime})=4$ which in turn implies that $(N:N^{\\prime})=2^{m+1}$ , as desired. ", "page_idx": 4}, {"type": "text", "text": "Finally, assume $d(G^{\\prime})\\geq3$ . Then $d(G^{\\prime}/G_{5})=3$ . Moreover there exists an exact sequence ", "page_idx": 4}, {"type": "equation", "text": "$$\nN/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and thus $\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\\,=\\,1$ which we now assume. $N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle$ and so, arguing as above, we have $N^{\\prime}=$ $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle$ , where $\\eta_{j}\\;\\in\\;G_{j}$ . But $N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=$ $\\langle c_{2}^{4}\\rangle$ . Therefore, $N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle$ . From this we see that $(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8$ and thus $(N:N^{\\prime})=2^{m+2}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Now suppose that $N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . ", "page_idx": 4}, {"type": "text", "text": "This establishes the proposition. ", "page_idx": 4}, {"type": "text", "text": "3. Number Theoretic Preliminaries ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\\operatorname{Gal}(M/k)\\simeq D_{4}$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. R´edei and Reichardt [12] proved this for $k=\\mathbb{Q}$ ; the general case is analogous. ", "page_idx": 4}, {"type": "text", "text": "We shall make extensive use of the class number formula for extensions of type $(2,2)$ : ", "page_idx": 4}, {"type": "text", "text": "Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then ", "page_idx": 4}, {"type": "equation", "text": "$$\nh(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\\kappa$ is the $\\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\\upsilon=0$ except when $K\\subseteq k(\\sqrt{E_{k}}\\,)$ , where $\\upsilon=1$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. See [10]. ", "page_idx": 4}]
|
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [{"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text"}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text"}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R´edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 92, 372, 101], "spans": [{"bbox": [239, 92, 372, 101], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [475, 689, 487, 700], "spans": [{"bbox": [475, 689, 487, 700], "score": 0.9910128712654114, "content": "口", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "5", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 447, 486, 458], "lines": [{"bbox": [475, 450, 487, 460], "spans": [{"bbox": [475, 450, 487, 460], "score": 0.9908492565155029, "content": "口", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [], "index": 0, "bbox_fs": [127, 115, 426, 126], "lines_deleted": true}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [126, 174, 487, 210]}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [137, 209, 261, 222]}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7, "bbox_fs": [126, 220, 487, 259]}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11, "bbox_fs": [124, 256, 487, 318]}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [125, 318, 487, 343]}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19, "bbox_fs": [125, 363, 487, 426]}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [124, 424, 487, 450]}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [137, 448, 279, 460]}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27, "bbox_fs": [126, 492, 487, 528]}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R´edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [126, 534, 486, 547]}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 565, 486, 591]}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5, "bbox_fs": [125, 597, 486, 621]}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [126, 646, 487, 682]}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38, "bbox_fs": [126, 687, 194, 702]}]}
|
[{"type": "text", "bbox": [125, 111, 427, 124], "content": "", "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "content": "", "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "content": "Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or .", "index": 2}, {"type": "text", "bbox": [135, 209, 261, 220], "content": "Suppose that .", "index": 3}, {"type": "text", "bbox": [125, 220, 487, 256], "content": "First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired.", "index": 4}, {"type": "text", "bbox": [124, 256, 486, 316], "content": "Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired.", "index": 5}, {"type": "text", "bbox": [125, 316, 486, 340], "content": "Finally, assume . Then . Moreover there exists an exact sequence", "index": 6}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "content": "", "index": 7}, {"type": "text", "bbox": [124, 362, 487, 422], "content": "and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired.", "index": 8}, {"type": "text", "bbox": [126, 423, 486, 447], "content": "Now suppose that . Then the proof is essentially the same as above once we notice that mod .", "index": 9}, {"type": "text", "bbox": [137, 447, 281, 459], "content": "This establishes the proposition.", "index": 10}, {"type": "title", "bbox": [213, 471, 397, 484], "content": "3. Number Theoretic Preliminaries", "index": 11}, {"type": "text", "bbox": [125, 489, 486, 526], "content": "Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and .", "index": 12}, {"type": "text", "bbox": [125, 532, 486, 545], "content": "Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.", "index": 13}, {"type": "text", "bbox": [125, 564, 487, 589], "content": "We shall make extensive use of the class number formula for extensions of type :", "index": 14}, {"type": "text", "bbox": [125, 595, 486, 619], "content": "Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then", "index": 15}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "content": "", "index": 16}, {"type": "text", "bbox": [125, 644, 486, 681], "content": "where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where .", "index": 17}, {"type": "text", "bbox": [126, 687, 192, 700], "content": "Proof. See [10].", "index": 18}]
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[{"bbox": [126, 174, 485, 187], "content": "Proof. Without loss of generality we assume that is metabelian. Let ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 186, 487, 199], "content": "where mod . Also let and (without loss", "parent_index": 2, "line_index": 1}, {"bbox": [126, 198, 359, 210], "content": "of generality). Then or .", "parent_index": 2, "line_index": 2}, {"bbox": [137, 209, 261, 222], "content": "Suppose that .", "parent_index": 3, "line_index": 0}, {"bbox": [137, 220, 487, 235], "content": "First assume . Then and thus . But", "parent_index": 4, "line_index": 0}, {"bbox": [126, 234, 486, 246], "content": "for some (cf. Lemma of [1]). Hence, , and so", "parent_index": 4, "line_index": 1}, {"bbox": [126, 244, 438, 259], "content": ". Since , we get as desired.", "parent_index": 4, "line_index": 2}, {"bbox": [136, 256, 487, 270], "content": "Next, assume that . Then by Lemma 1. Notice", "parent_index": 5, "line_index": 0}, {"bbox": [124, 268, 487, 284], "content": "that and where for . Hence", "parent_index": 5, "line_index": 1}, {"bbox": [126, 280, 487, 295], "content": "and so . But then by", "parent_index": 5, "line_index": 2}, {"bbox": [124, 293, 485, 307], "content": "Lemma 3. Therefore, by [5], . But notice that . Thus", "parent_index": 5, "line_index": 3}, {"bbox": [124, 304, 462, 318], "content": "and so which in turn implies that , as desired.", "parent_index": 5, "line_index": 4}, {"bbox": [137, 318, 487, 330], "content": "Finally, assume . Then . Moreover there exists an exact", "parent_index": 6, "line_index": 0}, {"bbox": [125, 331, 167, 343], "content": "sequence", "parent_index": 6, "line_index": 1}, {"bbox": [125, 363, 485, 378], "content": "and thus . Hence it suffices to prove the result for", "parent_index": 8, "line_index": 0}, {"bbox": [126, 376, 487, 390], "content": "which we now assume. and so, arguing as above, we have", "parent_index": 8, "line_index": 1}, {"bbox": [126, 388, 485, 402], "content": ", where . But", "parent_index": 8, "line_index": 2}, {"bbox": [126, 399, 487, 415], "content": ". Therefore, . From this we see that and thus", "parent_index": 8, "line_index": 3}, {"bbox": [126, 410, 250, 426], "content": "as desired.", "parent_index": 8, "line_index": 4}, {"bbox": [136, 424, 487, 437], "content": "Now suppose that . Then the proof is essentially the same as", "parent_index": 9, "line_index": 0}, {"bbox": [124, 434, 355, 450], "content": "above once we notice that mod .", "parent_index": 9, "line_index": 1}, {"bbox": [137, 448, 279, 460], "content": "This establishes the proposition.", "parent_index": 10, "line_index": 0}, {"bbox": [214, 474, 397, 484], "content": "3. Number Theoretic Preliminaries", "parent_index": 11, "line_index": 0}, {"bbox": [126, 492, 486, 505], "content": "Proposition 2. Let be a quadratic extension, and assume that the class num-", "parent_index": 12, "line_index": 0}, {"bbox": [126, 502, 487, 517], "content": "ber of , , is odd. If has an unramified cyclic extension M of order 4, then", "parent_index": 12, "line_index": 1}, {"bbox": [126, 515, 288, 528], "content": "is normal and .", "parent_index": 12, "line_index": 2}, {"bbox": [126, 534, 486, 547], "content": "Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.", "parent_index": 13, "line_index": 0}, {"bbox": [137, 565, 486, 579], "content": "We shall make extensive use of the class number formula for extensions of type", "parent_index": 14, "line_index": 0}, {"bbox": [126, 578, 153, 591], "content": ":", "parent_index": 14, "line_index": 1}, {"bbox": [125, 597, 486, 609], "content": "Proposition 3. Let be a normal quartic extension with Galois group of type", "parent_index": 15, "line_index": 0}, {"bbox": [126, 610, 437, 621], "content": ", and let ) denote the quadratic subextensions. Then", "parent_index": 15, "line_index": 1}, {"bbox": [127, 646, 487, 658], "content": "where denotes the unit index of is the unit group", "parent_index": 17, "line_index": 0}, {"bbox": [126, 658, 487, 670], "content": "of ), is the number of infinite primes in that ramify in , is the -rank", "parent_index": 17, "line_index": 1}, {"bbox": [126, 670, 466, 682], "content": "of the unit group of , and except when , where .", "parent_index": 17, "line_index": 2}, {"bbox": [126, 687, 194, 702], "content": "Proof. See [10].", "parent_index": 18, "line_index": 0}]
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[]
|
[{"bbox": [219, 129, 391, 169], "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "parent_index": 1, "subtype": "interline"}, {"bbox": [344, 176, 352, 183], "content": "G", "parent_index": 2, "subtype": "inline"}, {"bbox": [439, 175, 482, 186], "content": "G=\\langle a,b\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [155, 186, 213, 195], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "parent_index": 2, "subtype": "inline"}, {"bbox": [238, 188, 250, 197], "content": "G_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [295, 187, 345, 198], "content": "H=\\langle b,G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [369, 187, 424, 198], "content": "K=\\langle a b,G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 199, 276, 210], "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [292, 199, 356, 210], "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [198, 211, 258, 222], "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "parent_index": 3, "subtype": "inline"}, {"bbox": [195, 223, 236, 234], "content": "d(G^{\\prime})=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [268, 223, 308, 234], "content": "G^{\\prime}=\\langle c_{2}\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [350, 222, 416, 234], "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [442, 223, 487, 234], "content": "[a b^{2},c_{2}]=", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 235, 144, 246], "content": "c_{2}^{2}\\eta_{4}", "parent_index": 4, "subtype": "inline"}, {"bbox": [191, 235, 261, 246], "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [323, 236, 329, 243], "content": "^{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [403, 235, 448, 246], "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 247, 183, 258], "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [216, 247, 288, 258], "content": "(N:G^{\\prime})=2^{m-1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [325, 247, 388, 258], "content": "(N:N^{\\prime})=2^{m}", "parent_index": 4, "subtype": "inline"}, {"bbox": [225, 259, 271, 270], "content": "d(G^{\\prime})\\;=\\;2", "parent_index": 5, "subtype": "inline"}, {"bbox": [309, 258, 385, 270], "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [149, 270, 218, 282], "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [244, 270, 314, 282], "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [349, 272, 387, 282], "content": "\\eta_{j}~\\in~G_{j}", "parent_index": 5, "subtype": "inline"}, {"bbox": [409, 272, 447, 281], "content": "j~=~4,5", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 283, 218, 294], "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [253, 282, 326, 294], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [376, 282, 471, 294], "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 295, 286, 304], "content": "N^{\\prime}\\supseteq G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [361, 296, 399, 304], "content": "N_{3}\\subseteq G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [430, 295, 485, 306], "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [157, 307, 214, 317], "content": "(G^{\\prime}:N^{\\prime})=4", "parent_index": 5, "subtype": "inline"}, {"bbox": [335, 307, 408, 317], "content": "(N:N^{\\prime})=2^{m+1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [208, 319, 250, 329], "content": "d(G^{\\prime})\\geq3", "parent_index": 6, "subtype": "inline"}, {"bbox": [284, 319, 343, 329], "content": "d(G^{\\prime}/G_{5})=3", "parent_index": 6, "subtype": "inline"}, {"bbox": [230, 348, 380, 359], "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "parent_index": 7, "subtype": "interline"}, {"bbox": [169, 366, 263, 377], "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [451, 367, 485, 376], "content": "G_{5}\\,=\\,1", "parent_index": 8, "subtype": "inline"}, {"bbox": [227, 378, 311, 389], "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [462, 378, 487, 388], "content": "N^{\\prime}=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 389, 287, 401], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [325, 391, 361, 401], "content": "\\eta_{j}\\;\\in\\;G_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [393, 389, 485, 401], "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 402, 142, 412], "content": "\\langle c_{2}^{4}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [201, 401, 268, 412], "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [379, 402, 442, 412], "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 414, 199, 425], "content": "(N:N^{\\prime})=2^{m+2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [222, 425, 290, 437], "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [243, 437, 312, 448], "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "parent_index": 9, "subtype": "inline"}, {"bbox": [337, 439, 349, 448], "content": "G_{5}", "parent_index": 9, "subtype": "inline"}, {"bbox": [218, 493, 238, 503], "content": "K/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [154, 505, 160, 513], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [166, 505, 185, 515], "content": "h(k)", "parent_index": 12, "subtype": "inline"}, {"bbox": [236, 505, 245, 513], "content": "K", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 516, 147, 527], "content": "M/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [215, 516, 285, 527], "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "parent_index": 12, "subtype": "inline"}, {"bbox": [329, 536, 356, 545], "content": "k=\\mathbb{Q}", "parent_index": 13, "subtype": "inline"}, {"bbox": [126, 579, 148, 590], "content": "(2,2)", "parent_index": 14, "subtype": "inline"}, {"bbox": [219, 597, 239, 609], "content": "K/k", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 610, 148, 621], "content": "(2,2)", "parent_index": 15, "subtype": "inline"}, {"bbox": [188, 610, 198, 621], "content": "k_{j}", "parent_index": 15, "subtype": "inline"}, {"bbox": [202, 610, 248, 621], "content": "(j=1,2,3)", "parent_index": 15, "subtype": "inline"}, {"bbox": [203, 626, 404, 639], "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "parent_index": 16, "subtype": "interline"}, {"bbox": [154, 647, 255, 658], "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "parent_index": 17, "subtype": "inline"}, {"bbox": [371, 647, 390, 658], "content": "K/k", "parent_index": 17, "subtype": "inline"}, {"bbox": [396, 647, 409, 658], "content": "(E_{j}", "parent_index": 17, "subtype": "inline"}, {"bbox": [137, 659, 147, 670], "content": "k_{j}", "parent_index": 17, "subtype": "inline"}, {"bbox": [158, 659, 163, 667], "content": "d", "parent_index": 17, "subtype": "inline"}, {"bbox": [320, 658, 326, 667], "content": "k", "parent_index": 17, "subtype": "inline"}, {"bbox": [393, 659, 413, 669], "content": "K/k", "parent_index": 17, "subtype": "inline"}, {"bbox": [418, 660, 425, 667], "content": "\\kappa", "parent_index": 17, "subtype": "inline"}, {"bbox": [455, 659, 462, 667], "content": "\\mathbb{Z}", "parent_index": 17, "subtype": "inline"}, {"bbox": [202, 671, 215, 681], "content": "E_{k}", "parent_index": 17, "subtype": "inline"}, {"bbox": [230, 671, 236, 680], "content": "k", "parent_index": 17, "subtype": "inline"}, {"bbox": [261, 670, 286, 680], "content": "\\upsilon=0", "parent_index": 17, "subtype": "inline"}, {"bbox": [344, 670, 403, 682], "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "parent_index": 17, "subtype": "inline"}, {"bbox": [437, 672, 461, 680], "content": "\\upsilon=1", "parent_index": 17, "subtype": "inline"}]
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Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\mathrm{{Am}_{2}}$ its 2-Sylow subgroup.
Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by
$$
\#\operatorname{Am}(K/k)=h(k)\,{\frac{p^{t-1}}{(E:H)}},
$$
where $t$ is the number of primes (including those at $\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\times}$ . Moreover, $\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\nmid\#\operatorname{Am}(K/k)$ .
Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\operatorname{Am}(K/k)$ is defined by the exact sequence
$$
1\;\longrightarrow\;\mathrm{Am}(K/k)\;\longrightarrow\;\mathrm{Cl}(K)\;\longrightarrow\;\mathrm{Cl}(K)^{1-\sigma}\;\longrightarrow\;1,
$$
where $\sigma$ generates $\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\nmid\#\operatorname{Am}(K/k)$ is equivalent to $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$ . By induction we get $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$ , but since $(1-\sigma)^{p}\equiv0$ mod $p$ in the group ring $\mathbb{Z}[G]$ , this implies $\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$ . But then $\mathrm{Cl}_{p}(K)$ must be trivial. 口
We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\#\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$ .
We also need a result essentially due to G. Gras [4]:
Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and
$$
\operatorname{Cl}_{2}(K)\simeq\left\{\!\!\begin{array}{l l}{{(2,2)~o r~\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq3)~}}&{{i f\#\kappa_{K/k}=1,}}\\ {{\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq1)~}}&{{i f\#\kappa_{K/k}=2,}}\end{array}\!\!\right.
$$
where $\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ .
Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\mathrm{Cl}_{2}(K)=1$ , contrary to assumption.
Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\sigma$ be a generator of $G\,=\,\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$ . Moreover, let $\nu$ be the algebraic norm, that is, exponentiation by $1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$ . Then [4, Cor. 4.3] reads
Lemma 4. Suppose that $M^{\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\,=\,M$ and write $n=a(p-1)+b$ with integers $a\geq0$ and $0\,\leq\,b\,\leq\,p\,-\,2$ . If $\#\,M_{i+1}/M_{i}=p$ for $i=0,1,\dots,n-1$ , then $M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$ .
We claim that if $\kappa_{K/k}~=~2$ , then $M\ =\ \mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\,\in\,\mathrm{Cl}_{2}(K)$ , hence $M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$ . Moreover,
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<html><body>
<p data-bbox="124 111 487 149">Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. </p>
<p data-bbox="125 154 487 178">Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by </p>
<div class="equation" data-bbox="242 183 367 209">$$
\#\operatorname{Am}(K/k)=h(k)\,{\frac{p^{t-1}}{(E:H)}},
$$</div>
<p data-bbox="125 211 486 248">where $t$ is the number of primes (including those at $\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\times}$ . Moreover, $\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\nmid\#\operatorname{Am}(K/k)$ . </p>
<p data-bbox="126 252 487 277">Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\operatorname{Am}(K/k)$ is defined by the exact sequence </p>
<div class="equation" data-bbox="162 284 447 295">$$
1\;\longrightarrow\;\mathrm{Am}(K/k)\;\longrightarrow\;\mathrm{Cl}(K)\;\longrightarrow\;\mathrm{Cl}(K)^{1-\sigma}\;\longrightarrow\;1,
$$</div>
<p data-bbox="125 298 487 347">where $\sigma$ generates $\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\nmid\#\operatorname{Am}(K/k)$ is equivalent to $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$ . By induction we get $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$ , but since $(1-\sigma)^{p}\equiv0$ mod $p$ in the group ring $\mathbb{Z}[G]$ , this implies $\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$ . But then $\mathrm{Cl}_{p}(K)$ must be trivial. 口 </p>
<p data-bbox="125 353 486 389">We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\#\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$ . </p>
<p data-bbox="137 393 365 406">We also need a result essentially due to G. Gras [4]: </p>
<p data-bbox="125 411 486 436">Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and </p>
<div class="equation" data-bbox="189 441 421 473">$$
\operatorname{Cl}_{2}(K)\simeq\left\{\!\!\begin{array}{l l}{{(2,2)~o r~\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq3)~}}&{{i f\#\kappa_{K/k}=1,}}\\ {{\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq1)~}}&{{i f\#\kappa_{K/k}=2,}}\end{array}\!\!\right.
$$</div>
<p data-bbox="126 475 486 499">where $\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . </p>
<p data-bbox="125 504 486 541">Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. </p>
<p data-bbox="125 542 487 615">Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\sigma$ be a generator of $G\,=\,\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$ . Moreover, let $\nu$ be the algebraic norm, that is, exponentiation by $1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$ . Then [4, Cor. 4.3] reads </p>
<p data-bbox="124 619 487 658">Lemma 4. Suppose that $M^{\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\,=\,M$ and write $n=a(p-1)+b$ with integers $a\geq0$ and $0\,\leq\,b\,\leq\,p\,-\,2$ . If $\#\,M_{i+1}/M_{i}=p$ for $i=0,1,\dots,n-1$ , then $M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$ . </p>
<p data-bbox="124 662 487 701">We claim that if $\kappa_{K/k}~=~2$ , then $M\ =\ \mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\,\in\,\mathrm{Cl}_{2}(K)$ , hence $M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$ . Moreover, </p>
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0003244v1
| 5 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. ", "page_idx": 5}, {"type": "text", "text": "Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $t$ is the number of primes (including those at $\\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\\times}$ . Moreover, $\\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\\nmid\\#\\operatorname{Am}(K/k)$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\\operatorname{Am}(K/k)$ is defined by the exact sequence ", "page_idx": 5}, {"type": "equation", "text": "$$\n1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\sigma$ generates $\\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\\nmid\\#\\operatorname{Am}(K/k)$ is equivalent to $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}$ . By induction we get $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}$ , but since $(1-\\sigma)^{p}\\equiv0$ mod $p$ in the group ring $\\mathbb{Z}[G]$ , this implies $\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}$ . But then $\\mathrm{Cl}_{p}(K)$ must be trivial. 口 ", "page_idx": 5}, {"type": "text", "text": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\\#\\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)$ . ", "page_idx": 5}, {"type": "text", "text": "We also need a result essentially due to G. Gras [4]: ", "page_idx": 5}, {"type": "text", "text": "Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. ", "page_idx": 5}, {"type": "text", "text": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\\sigma$ be a generator of $G\\,=\\,\\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}$ . Moreover, let $\\nu$ be the algebraic norm, that is, exponentiation by $1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}$ . Then [4, Cor. 4.3] reads ", "page_idx": 5}, {"type": "text", "text": "Lemma 4. Suppose that $M^{\\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\\,=\\,M$ and write $n=a(p-1)+b$ with integers $a\\geq0$ and $0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2$ . If $\\#\\,M_{i+1}/M_{i}=p$ for $i=0,1,\\dots,n-1$ , then $M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}$ . ", "page_idx": 5}, {"type": "text", "text": "We claim that if $\\kappa_{K/k}~=~2$ , then $M\\ =\\ \\mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\\,\\in\\,\\mathrm{Cl}_{2}(K)$ , hence $M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1$ . Moreover, ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "口", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [124, 90, 131, 99], "lines": [{"bbox": [126, 93, 132, 101], "spans": [{"bbox": [126, 93, 132, 101], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [126, 114, 486, 151]}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [125, 156, 486, 181]}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "bbox_fs": [125, 213, 487, 250]}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 254, 487, 280]}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "口", "type": "text"}], "index": 15}], "index": 13.5, "bbox_fs": [124, 299, 487, 349]}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17, "bbox_fs": [125, 355, 486, 391]}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [137, 395, 364, 408]}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [126, 414, 487, 437]}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 475, 487, 500]}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [126, 507, 486, 545]}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5, "bbox_fs": [124, 542, 487, 618]}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35, "bbox_fs": [124, 622, 487, 660]}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38, "bbox_fs": [123, 663, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 487, 149], "content": "Another important result is the ambiguous class number formula. For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.", "index": 0}, {"type": "text", "bbox": [125, 154, 487, 178], "content": "Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by", "index": 1}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "content": "", "index": 2}, {"type": "text", "bbox": [125, 211, 486, 248], "content": "where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if .", "index": 3}, {"type": "text", "bbox": [126, 252, 487, 277], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence", "index": 4}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "content": "", "index": 5}, {"type": "text", "bbox": [125, 298, 487, 347], "content": "where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口", "index": 6}, {"type": "text", "bbox": [125, 353, 486, 389], "content": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where .", "index": 7}, {"type": "text", "bbox": [137, 393, 365, 406], "content": "We also need a result essentially due to G. Gras [4]:", "index": 8}, {"type": "text", "bbox": [125, 411, 486, 436], "content": "Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and", "index": 9}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "content": "", "index": 10}, {"type": "text", "bbox": [126, 475, 486, 499], "content": "where denotes the set of ideal classes of that become principal (capitulate) in .", "index": 11}, {"type": "text", "bbox": [125, 504, 486, 541], "content": "Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption.", "index": 12}, {"type": "text", "bbox": [125, 542, 487, 615], "content": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads", "index": 13}, {"type": "text", "bbox": [124, 619, 487, 658], "content": "Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then .", "index": 14}, {"type": "text", "bbox": [124, 662, 487, 701], "content": "We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,", "index": 15}]
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[{"bbox": [138, 114, 486, 126], "content": "Another important result is the ambiguous class number formula. For cyclic", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 485, 138], "content": "extensions , let denote the group of ideal classes in fixed by", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 151], "content": ", i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.", "parent_index": 0, "line_index": 2}, {"bbox": [125, 156, 486, 169], "content": "Proposition 4. Let be a cyclic extension of prime degree ; then the number", "parent_index": 1, "line_index": 0}, {"bbox": [126, 168, 291, 181], "content": "of ambiguous ideal classes is given by", "parent_index": 1, "line_index": 1}, {"bbox": [126, 213, 485, 225], "content": "where is the number of primes (including those at ) of that ramify in ,", "parent_index": 3, "line_index": 0}, {"bbox": [125, 225, 487, 237], "content": "is the unit group of , and is its subgroup consisting of norms of elements from", "parent_index": 3, "line_index": 1}, {"bbox": [126, 237, 401, 250], "content": ". Moreover, is trivial if and only if .", "parent_index": 3, "line_index": 2}, {"bbox": [126, 254, 487, 269], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "parent_index": 4, "line_index": 0}, {"bbox": [125, 266, 456, 280], "content": "(see e.g. Moriya [11]), note that is defined by the exact sequence", "parent_index": 4, "line_index": 1}, {"bbox": [126, 299, 485, 313], "content": "where generates . Taking -parts we see that is equiv-", "parent_index": 6, "line_index": 0}, {"bbox": [124, 311, 487, 327], "content": "alent to . By induction we get , but", "parent_index": 6, "line_index": 1}, {"bbox": [125, 324, 485, 338], "content": "since mod in the group ring , this implies .", "parent_index": 6, "line_index": 2}, {"bbox": [125, 336, 486, 349], "content": "But then must be trivial. 口", "parent_index": 6, "line_index": 3}, {"bbox": [125, 355, 486, 367], "content": "We make one further remark concerning the ambiguous class number formula that", "parent_index": 7, "line_index": 0}, {"bbox": [126, 367, 486, 379], "content": "will be useful below. If the class number is odd, then it is known that", "parent_index": 7, "line_index": 1}, {"bbox": [126, 379, 312, 391], "content": "where .", "parent_index": 7, "line_index": 2}, {"bbox": [137, 395, 364, 408], "content": "We also need a result essentially due to G. Gras [4]:", "parent_index": 8, "line_index": 0}, {"bbox": [126, 414, 487, 426], "content": "Proposition 5. Let be a quadratic extension of number fields and assume", "parent_index": 9, "line_index": 0}, {"bbox": [126, 426, 382, 437], "content": "that . Then is ramified and", "parent_index": 9, "line_index": 1}, {"bbox": [125, 475, 487, 492], "content": "where denotes the set of ideal classes of that become principal (capitulate)", "parent_index": 11, "line_index": 0}, {"bbox": [126, 489, 152, 500], "content": "in .", "parent_index": 11, "line_index": 1}, {"bbox": [127, 507, 485, 519], "content": "Proof. We first notice that is ramified. If the extension were unramified, then", "parent_index": 12, "line_index": 0}, {"bbox": [126, 518, 486, 531], "content": "would be the 2-class field of , and since is cyclic, it would follow that", "parent_index": 12, "line_index": 1}, {"bbox": [126, 530, 286, 545], "content": ", contrary to assumption.", "parent_index": 12, "line_index": 2}, {"bbox": [137, 542, 486, 555], "content": "Before we start with the rest of the proof, we cite the results of Gras that", "parent_index": 13, "line_index": 0}, {"bbox": [126, 555, 486, 568], "content": "we need (we could also give a slightly longer direct proof without referring to", "parent_index": 13, "line_index": 1}, {"bbox": [125, 565, 484, 580], "content": "his results). Let be a cyclic extension of prime power order , and let", "parent_index": 13, "line_index": 2}, {"bbox": [124, 578, 486, 592], "content": "be a generator of . For any -group on which acts we put", "parent_index": 13, "line_index": 3}, {"bbox": [126, 589, 487, 605], "content": ". Moreover, let be the algebraic norm, that is,", "parent_index": 13, "line_index": 4}, {"bbox": [124, 602, 429, 618], "content": "exponentiation by . Then [4, Cor. 4.3] reads", "parent_index": 13, "line_index": 5}, {"bbox": [124, 622, 487, 635], "content": "Lemma 4. Suppose that ; let be the smallest positive integer such that", "parent_index": 14, "line_index": 0}, {"bbox": [126, 634, 487, 647], "content": "and write with integers and . If", "parent_index": 14, "line_index": 1}, {"bbox": [125, 645, 472, 660], "content": "for , then .", "parent_index": 14, "line_index": 2}, {"bbox": [137, 663, 487, 677], "content": "We claim that if , then satisfies the assumptions of", "parent_index": 15, "line_index": 0}, {"bbox": [123, 676, 486, 690], "content": "Lemma 4: in fact, let denote the transfer of ideal classes. Then", "parent_index": 15, "line_index": 1}, {"bbox": [126, 689, 486, 702], "content": "for any ideal class , hence . Moreover,", "parent_index": 15, "line_index": 2}]
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[]
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[{"bbox": [175, 127, 195, 138], "content": "K/k", "parent_index": 0, "subtype": "inline"}, {"bbox": [219, 127, 262, 138], "content": "\\operatorname{Am}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [434, 128, 444, 135], "content": "K", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 139, 168, 150], "content": "\\operatorname{Gal}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [338, 140, 347, 147], "content": "K", "parent_index": 0, "subtype": "inline"}, {"bbox": [370, 140, 390, 149], "content": "\\mathrm{{Am}_{2}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [218, 158, 238, 168], "content": "K/k", "parent_index": 1, "subtype": "inline"}, {"bbox": [403, 161, 408, 167], "content": "p", "parent_index": 1, "subtype": "inline"}, {"bbox": [242, 183, 367, 209], "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "parent_index": 2, "subtype": "interline"}, {"bbox": [154, 215, 158, 222], "content": "t", "parent_index": 3, "subtype": "inline"}, {"bbox": [351, 217, 361, 222], "content": "\\infty", "parent_index": 3, "subtype": "inline"}, {"bbox": [379, 214, 385, 222], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [451, 214, 471, 225], "content": "K/k", "parent_index": 3, "subtype": "inline"}, {"bbox": [477, 215, 485, 222], "content": "E", "parent_index": 3, "subtype": "inline"}, {"bbox": [213, 227, 219, 234], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [245, 227, 254, 234], "content": "H", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 238, 142, 246], "content": "K^{\\times}", "parent_index": 3, "subtype": "inline"}, {"bbox": [196, 238, 228, 249], "content": "\\mathrm{Cl}_{p}(K)", "parent_index": 3, "subtype": "inline"}, {"bbox": [331, 237, 398, 249], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "parent_index": 3, "subtype": "inline"}, {"bbox": [268, 268, 311, 279], "content": "\\operatorname{Am}(K/k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [162, 284, 447, 295], "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "parent_index": 5, "subtype": "interline"}, {"bbox": [155, 304, 161, 309], "content": "\\sigma", "parent_index": 6, "subtype": "inline"}, {"bbox": [208, 301, 251, 312], "content": "\\operatorname{Gal}(K/k)", "parent_index": 6, "subtype": "inline"}, {"bbox": [291, 304, 297, 311], "content": "p", "parent_index": 6, "subtype": "inline"}, {"bbox": [378, 301, 445, 312], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "parent_index": 6, "subtype": "inline"}, {"bbox": [164, 313, 257, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "parent_index": 6, "subtype": "inline"}, {"bbox": [359, 313, 463, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [151, 325, 206, 336], "content": "(1-\\sigma)^{p}\\equiv0", "parent_index": 6, "subtype": "inline"}, {"bbox": [231, 329, 236, 336], "content": "p", "parent_index": 6, "subtype": "inline"}, {"bbox": [318, 326, 339, 336], "content": "\\mathbb{Z}[G]", "parent_index": 6, "subtype": "inline"}, {"bbox": [399, 326, 482, 336], "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "parent_index": 6, "subtype": "inline"}, {"bbox": [168, 338, 200, 348], "content": "\\mathrm{Cl}_{p}(K)", "parent_index": 6, "subtype": "inline"}, {"bbox": [324, 368, 343, 379], "content": "h(k)", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 380, 205, 390], "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "parent_index": 7, "subtype": "inline"}, {"bbox": [238, 380, 309, 390], "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "parent_index": 7, "subtype": "inline"}, {"bbox": [220, 415, 240, 425], "content": "K/k", "parent_index": 9, "subtype": "inline"}, {"bbox": [146, 427, 259, 437], "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "parent_index": 9, "subtype": "inline"}, {"bbox": [292, 426, 312, 437], "content": "K/k", "parent_index": 9, "subtype": "inline"}, {"bbox": [189, 441, 421, 473], "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "parent_index": 10, "subtype": "interline"}, {"bbox": [154, 481, 176, 489], "content": "\\kappa_{K/k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [329, 478, 335, 486], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [138, 491, 148, 498], "content": "K", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 508, 264, 518], "content": "K/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 520, 135, 528], "content": "K", "parent_index": 12, "subtype": "inline"}, {"bbox": [267, 520, 272, 528], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [324, 520, 352, 530], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 532, 176, 542], "content": "\\mathrm{Cl}_{2}(K)=1", "parent_index": 12, "subtype": "inline"}, {"bbox": [204, 568, 224, 578], "content": "K/k", "parent_index": 13, "subtype": "inline"}, {"bbox": [425, 568, 435, 577], "content": "p^{r}", "parent_index": 13, "subtype": "inline"}, {"bbox": [478, 571, 484, 576], "content": "\\sigma", "parent_index": 13, "subtype": "inline"}, {"bbox": [207, 579, 275, 590], "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "parent_index": 13, "subtype": "inline"}, {"bbox": [323, 583, 329, 589], "content": "p", "parent_index": 13, "subtype": "inline"}, {"bbox": [361, 580, 372, 587], "content": "M", "parent_index": 13, "subtype": "inline"}, {"bbox": [420, 580, 428, 588], "content": "G", "parent_index": 13, "subtype": "inline"}, {"bbox": [126, 590, 265, 604], "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "parent_index": 13, "subtype": "inline"}, {"bbox": [338, 596, 344, 601], "content": "\\nu", "parent_index": 13, "subtype": "inline"}, {"bbox": [207, 604, 314, 614], "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [239, 623, 275, 632], "content": "M^{\\nu}=1", "parent_index": 14, "subtype": "inline"}, {"bbox": [297, 627, 303, 631], "content": "n", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 635, 168, 645], "content": "M_{n}\\,=\\,M", "parent_index": 14, "subtype": "inline"}, {"bbox": [217, 635, 292, 646], "content": "n=a(p-1)+b", "parent_index": 14, "subtype": "inline"}, {"bbox": [356, 636, 381, 644], "content": "a\\geq0", "parent_index": 14, "subtype": "inline"}, {"bbox": [405, 636, 469, 645], "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "parent_index": 14, "subtype": "inline"}, {"bbox": [125, 646, 195, 658], "content": "\\#\\,M_{i+1}/M_{i}=p", "parent_index": 14, "subtype": "inline"}, {"bbox": [214, 648, 293, 657], "content": "i=0,1,\\dots,n-1", "parent_index": 14, "subtype": "inline"}, {"bbox": [321, 646, 468, 658], "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "parent_index": 14, "subtype": "inline"}, {"bbox": [218, 665, 264, 677], "content": "\\kappa_{K/k}~=~2", "parent_index": 15, "subtype": "inline"}, {"bbox": [296, 666, 357, 676], "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "parent_index": 15, "subtype": "inline"}, {"bbox": [223, 678, 266, 688], "content": "j=j_{k\\rightarrow K}", "parent_index": 15, "subtype": "inline"}, {"bbox": [454, 677, 486, 687], "content": "c^{1+\\sigma}=", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 690, 167, 702], "content": "j(N_{K/k}c)", "parent_index": 15, "subtype": "inline"}, {"bbox": [254, 689, 305, 701], "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "parent_index": 15, "subtype": "inline"}, {"bbox": [340, 690, 432, 701], "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "parent_index": 15, "subtype": "inline"}]
|
[]
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$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$ , that is, the 2-class group is cyclic.
The second result of Gras that we need is [4, Prop. 4.3]
Lemma 5. Suppose that $M^{\nu}\ne1$ but assume the other conditions in Lemma 4. Then $n\geq2$ and
$$
M\simeq\left\{\!\!\begin{array}{l l}{(\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n<p;}\\ {(\mathbb{Z}/p\mathbb{Z})^{p}\ \mathrm{or}\ (\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n=p;}\\ {(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}}&{\mathrm{if}\ n>p.}\end{array}\right.
$$
If $\kappa_{K/k}=1$ , then this lemma shows that $\mathrm{Cl}_{2}(K)$ is either cyclic of order $\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\mathrm{Cl}_{2}(K)\simeq$ $\mathbb{Z}/4\mathbb{Z}$ cannot occur here.
Now assume that $\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto, and using $\kappa_{K/k}=1$ once more we find $C^{1+\sigma}=c$ , where $c$ is the nontrivial ideal class from $\mathrm{Cl_{2}}(k)$ . On the other hand, $c\in\mathrm{Cl}_{2}(k)$ still has order 2 in $\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\sigma}$ . But this implies that $C^{\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\#\operatorname{Am}_{2}(K/k)=2$ . 口
# 4. Arithmetic of some Dihedral Extensions
In this section we study the arithmetic of some dihedral extensions $L/\mathbb{Q}$ , that is, normal extensions $L$ of $\mathbb{Q}$ with Galois group $\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\langle\tau,\sigma|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$ . Now consider the following diagrams (Galois correspondence):
In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\prime}$ respectively. Finally, let $\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ .
The following remark will be used several times: if $K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{1}$ , then $k_{2}=\mathbb{Q}({\sqrt{a}}\,)$ , where $a=\alpha\alpha^{\prime}$ is the norm of $\alpha$ . To see this, let $\gamma=\sqrt{\alpha}$ ; then $\gamma^{\tau}\,=\,\gamma$ , since $\gamma\ \in\ K_{1}$ . Clearly $\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$ and hence fixed by $\sigma^{2}$ . Furthermore,
$$
(\gamma^{1+\sigma})^{\sigma\tau}=\gamma^{\sigma\tau+\sigma^{2}\tau}=\gamma^{\tau\sigma^{3}+\tau\sigma^{2}}=(\gamma^{\tau})^{\sigma^{3}+\sigma^{2}}=\gamma^{\sigma^{3}+\sigma^{2}}=\gamma^{(1+\sigma)\sigma^{2}}=\gamma^{1+\sigma},
$$
implying that ${\sqrt{a}}\ \in\ k_{2}$ . Finally notice that ${\sqrt{a}}\ \not\in\ \mathbb{Q}$ , since otherwise $\sqrt{\alpha^{\prime}}\,=$ ${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$ implying that $K_{1}/\mathbb{Q}$ is normal, which is not the case.
|
<html><body>
<p data-bbox="125 111 487 160">$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$ , that is, the 2-class group is cyclic. </p>
<p data-bbox="136 160 380 172">The second result of Gras that we need is [4, Prop. 4.3] </p>
<p data-bbox="124 177 486 201">Lemma 5. Suppose that $M^{\nu}\ne1$ but assume the other conditions in Lemma 4. Then $n\geq2$ and </p>
<div class="equation" data-bbox="191 205 420 251">$$
M\simeq\left\{\!\!\begin{array}{l l}{(\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n<p;}\\ {(\mathbb{Z}/p\mathbb{Z})^{p}\ \mathrm{or}\ (\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n=p;}\\ {(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}}&{\mathrm{if}\ n>p.}\end{array}\right.
$$</div>
<p data-bbox="124 253 487 314">If $\kappa_{K/k}=1$ , then this lemma shows that $\mathrm{Cl}_{2}(K)$ is either cyclic of order $\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\mathrm{Cl}_{2}(K)\simeq$ $\mathbb{Z}/4\mathbb{Z}$ cannot occur here. </p>
<p data-bbox="125 314 486 387">Now assume that $\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto, and using $\kappa_{K/k}=1$ once more we find $C^{1+\sigma}=c$ , where $c$ is the nontrivial ideal class from $\mathrm{Cl_{2}}(k)$ . On the other hand, $c\in\mathrm{Cl}_{2}(k)$ still has order 2 in $\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\sigma}$ . But this implies that $C^{\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\#\operatorname{Am}_{2}(K/k)=2$ . 口 </p>
<h1 data-bbox="193 393 418 406">4. Arithmetic of some Dihedral Extensions </h1>
<p data-bbox="124 412 486 460">In this section we study the arithmetic of some dihedral extensions $L/\mathbb{Q}$ , that is, normal extensions $L$ of $\mathbb{Q}$ with Galois group $\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\langle\tau,\sigma|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$ . Now consider the following diagrams (Galois correspondence): </p>
<div class="image" data-bbox="131 468 478 554"><img data-bbox="131 468 478 554"/></div>
<p data-bbox="125 556 487 605">In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\prime}$ respectively. Finally, let $\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . </p>
<p data-bbox="125 606 486 653">The following remark will be used several times: if $K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{1}$ , then $k_{2}=\mathbb{Q}({\sqrt{a}}\,)$ , where $a=\alpha\alpha^{\prime}$ is the norm of $\alpha$ . To see this, let $\gamma=\sqrt{\alpha}$ ; then $\gamma^{\tau}\,=\,\gamma$ , since $\gamma\ \in\ K_{1}$ . Clearly $\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$ and hence fixed by $\sigma^{2}$ . Furthermore, </p>
<div class="equation" data-bbox="142 658 466 672">$$
(\gamma^{1+\sigma})^{\sigma\tau}=\gamma^{\sigma\tau+\sigma^{2}\tau}=\gamma^{\tau\sigma^{3}+\tau\sigma^{2}}=(\gamma^{\tau})^{\sigma^{3}+\sigma^{2}}=\gamma^{\sigma^{3}+\sigma^{2}}=\gamma^{(1+\sigma)\sigma^{2}}=\gamma^{1+\sigma},
$$</div>
<p data-bbox="124 674 486 700">implying that ${\sqrt{a}}\ \in\ k_{2}$ . Finally notice that ${\sqrt{a}}\ \not\in\ \mathbb{Q}$ , since otherwise $\sqrt{\alpha^{\prime}}\,=$ ${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$ implying that $K_{1}/\mathbb{Q}$ is normal, which is not the case. </p>
</body></html>
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0003244v1
| 6 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "$M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}$ , that is, the 2-class group is cyclic. ", "page_idx": 6}, {"type": "text", "text": "The second result of Gras that we need is [4, Prop. 4.3] ", "page_idx": 6}, {"type": "text", "text": "Lemma 5. Suppose that $M^{\\nu}\\ne1$ but assume the other conditions in Lemma 4. Then $n\\geq2$ and ", "page_idx": 6}, {"type": "equation", "text": "$$\nM\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "If $\\kappa_{K/k}=1$ , then this lemma shows that $\\mathrm{Cl}_{2}(K)$ is either cyclic of order $\\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\\mathrm{Cl}_{2}(K)\\simeq$ $\\mathbb{Z}/4\\mathbb{Z}$ cannot occur here. ", "page_idx": 6}, {"type": "text", "text": "Now assume that $\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto, and using $\\kappa_{K/k}=1$ once more we find $C^{1+\\sigma}=c$ , where $c$ is the nontrivial ideal class from $\\mathrm{Cl_{2}}(k)$ . On the other hand, $c\\in\\mathrm{Cl}_{2}(k)$ still has order 2 in $\\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\\sigma}$ . But this implies that $C^{\\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\\#\\operatorname{Am}_{2}(K/k)=2$ . 口 ", "page_idx": 6}, {"type": "text", "text": "4. Arithmetic of some Dihedral Extensions ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "In this section we study the arithmetic of some dihedral extensions $L/\\mathbb{Q}$ , that is, normal extensions $L$ of $\\mathbb{Q}$ with Galois group $\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle$ . Now consider the following diagrams (Galois correspondence): ", "page_idx": 6}, {"type": "image", "img_path": "images/110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\\prime}$ respectively. Finally, let $\\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . ", "page_idx": 6}, {"type": "text", "text": "The following remark will be used several times: if $K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{1}$ , then $k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)$ , where $a=\\alpha\\alpha^{\\prime}$ is the norm of $\\alpha$ . To see this, let $\\gamma=\\sqrt{\\alpha}$ ; then $\\gamma^{\\tau}\\,=\\,\\gamma$ , since $\\gamma\\ \\in\\ K_{1}$ . Clearly $\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K$ and hence fixed by $\\sigma^{2}$ . Furthermore, ", "page_idx": 6}, {"type": "equation", "text": "$$\n(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "implying that ${\\sqrt{a}}\\ \\in\\ k_{2}$ . Finally notice that ${\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}$ , since otherwise $\\sqrt{\\alpha^{\\prime}}\\,=$ ${\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}$ implying that $K_{1}/\\mathbb{Q}$ is normal, which is not the case. ", "page_idx": 6}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "口", "type": "text"}], "index": 18}], "index": 15.5}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [481, 94, 485, 100], "spans": [{"bbox": [481, 94, 485, 100], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [126, 114, 487, 162]}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4, "bbox_fs": [137, 162, 381, 173]}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [125, 180, 486, 204]}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10, "bbox_fs": [124, 255, 487, 317]}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "口", "type": "text"}], "index": 18}], "index": 15.5, "bbox_fs": [124, 315, 487, 388]}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5, "bbox_fs": [125, 413, 487, 462]}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5, "bbox_fs": [124, 559, 486, 608]}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5, "bbox_fs": [124, 605, 487, 655]}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5, "bbox_fs": [125, 676, 486, 702]}]}
|
[{"type": "text", "bbox": [125, 111, 487, 160], "content": "in our case, hence has order 2. Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic.", "index": 0}, {"type": "text", "bbox": [136, 160, 380, 172], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "index": 1}, {"type": "text", "bbox": [124, 177, 486, 201], "content": "Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and", "index": 2}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "content": "", "index": 3}, {"type": "text", "bbox": [124, 253, 487, 314], "content": "If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here.", "index": 4}, {"type": "text", "bbox": [125, 314, 486, 387], "content": "Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口", "index": 5}, {"type": "title", "bbox": [193, 393, 418, 406], "content": "4. Arithmetic of some Dihedral Extensions", "index": 6}, {"type": "text", "bbox": [124, 412, 486, 460], "content": "In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence):", "index": 7}, {"type": "image", "bbox": [131, 468, 478, 554], "content": "", "index": 8}, {"type": "text", "bbox": [125, 556, 487, 605], "content": "In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for .", "index": 9}, {"type": "text", "bbox": [125, 606, 486, 653], "content": "The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore,", "index": 10}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "content": "", "index": 11}, {"type": "text", "bbox": [124, 674, 486, 700], "content": "implying that . Finally notice that , since otherwise implying that is normal, which is not the case.", "index": 12}]
|
[{"bbox": [126, 114, 487, 126], "content": "in our case, hence has order 2. Since the orders of", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 486, 138], "content": "decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 150], "content": "# for all . Since and when , Lemma 4 now", "parent_index": 0, "line_index": 2}, {"bbox": [126, 150, 410, 162], "content": "implies that , that is, the 2-class group is cyclic.", "parent_index": 0, "line_index": 3}, {"bbox": [137, 162, 381, 173], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "parent_index": 1, "line_index": 0}, {"bbox": [125, 180, 486, 192], "content": "Lemma 5. Suppose that but assume the other conditions in Lemma 4.", "parent_index": 2, "line_index": 0}, {"bbox": [127, 192, 199, 204], "content": "Then and", "parent_index": 2, "line_index": 1}, {"bbox": [136, 255, 485, 269], "content": "If , then this lemma shows that is either cyclic of order", "parent_index": 4, "line_index": 0}, {"bbox": [125, 268, 485, 280], "content": "or of type . (Notice that the hypothesis of the lemma is satisfied since", "parent_index": 4, "line_index": 1}, {"bbox": [124, 279, 487, 294], "content": "is ramified implying that the norm is onto; and so the", "parent_index": 4, "line_index": 2}, {"bbox": [125, 292, 486, 304], "content": "argument above this lemma applies.) It remains to show that the case", "parent_index": 4, "line_index": 3}, {"bbox": [126, 303, 234, 317], "content": "cannot occur here.", "parent_index": 4, "line_index": 4}, {"bbox": [135, 315, 487, 329], "content": "Now assume that ; since is ramified, the norm", "parent_index": 5, "line_index": 0}, {"bbox": [126, 327, 487, 342], "content": "is onto, and using once more we find ,", "parent_index": 5, "line_index": 1}, {"bbox": [126, 339, 486, 352], "content": "where is the nontrivial ideal class from . On the other hand, still", "parent_index": 5, "line_index": 2}, {"bbox": [124, 350, 487, 365], "content": "has order 2 in , hence we must also have . But this implies that", "parent_index": 5, "line_index": 3}, {"bbox": [126, 363, 485, 376], "content": ", i.e. that each ideal class in is ambiguous, contradicting our assumption", "parent_index": 5, "line_index": 4}, {"bbox": [126, 376, 487, 388], "content": "that . 口", "parent_index": 5, "line_index": 5}, {"bbox": [193, 396, 418, 407], "content": "4. Arithmetic of some Dihedral Extensions", "parent_index": 6, "line_index": 0}, {"bbox": [137, 413, 485, 426], "content": "In this section we study the arithmetic of some dihedral extensions , that is,", "parent_index": 7, "line_index": 0}, {"bbox": [125, 425, 487, 439], "content": "normal extensions of with Galois group , the dihedral group", "parent_index": 7, "line_index": 1}, {"bbox": [125, 437, 487, 451], "content": "of order 8. Hence may be presented as . Now", "parent_index": 7, "line_index": 2}, {"bbox": [126, 450, 375, 462], "content": "consider the following diagrams (Galois correspondence):", "parent_index": 7, "line_index": 3}, {"bbox": [137, 559, 486, 571], "content": "In this situation, we let and denote", "parent_index": 9, "line_index": 0}, {"bbox": [126, 571, 486, 584], "content": "the unit indices of the bicyclic extensions and , where and are", "parent_index": 9, "line_index": 1}, {"bbox": [126, 583, 486, 596], "content": "the unit groups in and respectively. Finally, let denote the kernel of the", "parent_index": 9, "line_index": 2}, {"bbox": [124, 594, 408, 608], "content": "transfer of ideal classes for .", "parent_index": 9, "line_index": 3}, {"bbox": [137, 605, 487, 619], "content": "The following remark will be used several times: if for some", "parent_index": 10, "line_index": 0}, {"bbox": [126, 618, 485, 632], "content": ", then , where is the norm of . To see this, let ;", "parent_index": 10, "line_index": 1}, {"bbox": [124, 630, 487, 644], "content": "then , since . Clearly and hence fixed by .", "parent_index": 10, "line_index": 2}, {"bbox": [125, 642, 185, 655], "content": "Furthermore,", "parent_index": 10, "line_index": 3}, {"bbox": [125, 676, 486, 689], "content": "implying that . Finally notice that , since otherwise", "parent_index": 12, "line_index": 0}, {"bbox": [126, 689, 422, 702], "content": "implying that is normal, which is not the case.", "parent_index": 12, "line_index": 1}]
|
[{"bbox": [131, 468, 478, 554], "content": "", "parent_index": 8, "subtype": "body"}]
|
[{"bbox": [126, 115, 206, 126], "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [298, 115, 330, 126], "content": "M_{1}/M_{0}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 167, 138], "content": "M_{i+1}/M_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [246, 128, 252, 135], "content": "^{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [267, 128, 271, 135], "content": "i", "parent_index": 0, "subtype": "inline"}, {"bbox": [133, 139, 196, 150], "content": ":M_{i+1}/M_{i}=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [230, 140, 254, 147], "content": "i<n", "parent_index": 0, "subtype": "inline"}, {"bbox": [288, 142, 314, 147], "content": "a=n", "parent_index": 0, "subtype": "inline"}, {"bbox": [338, 140, 361, 147], "content": "b=0", "parent_index": 0, "subtype": "inline"}, {"bbox": [392, 140, 416, 149], "content": "p=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [181, 151, 255, 162], "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "parent_index": 0, "subtype": "inline"}, {"bbox": [240, 181, 277, 191], "content": "M^{\\nu}\\ne1", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 194, 176, 203], "content": "n\\geq2", "parent_index": 2, "subtype": "inline"}, {"bbox": [191, 205, 420, 251], "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "parent_index": 3, "subtype": "interline"}, {"bbox": [149, 258, 191, 268], "content": "\\kappa_{K/k}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [326, 257, 357, 267], "content": "\\mathrm{Cl}_{2}(K)", "parent_index": 4, "subtype": "inline"}, {"bbox": [468, 258, 485, 266], "content": "\\geq4", "parent_index": 4, "subtype": "inline"}, {"bbox": [173, 269, 196, 280], "content": "(2,2)", "parent_index": 4, "subtype": "inline"}, {"bbox": [465, 269, 485, 280], "content": "K/k", "parent_index": 4, "subtype": "inline"}, {"bbox": [283, 281, 398, 292], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [442, 293, 486, 303], "content": "\\mathrm{Cl}_{2}(K)\\simeq", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 305, 150, 316], "content": "\\mathbb{Z}/4\\mathbb{Z}", "parent_index": 4, "subtype": "inline"}, {"bbox": [221, 317, 330, 327], "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "parent_index": 5, "subtype": "inline"}, {"bbox": [363, 317, 383, 327], "content": "K/k", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 329, 239, 340], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [319, 330, 359, 340], "content": "\\kappa_{K/k}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [441, 328, 482, 337], "content": "C^{1+\\sigma}=c", "parent_index": 5, "subtype": "inline"}, {"bbox": [154, 344, 159, 349], "content": "c", "parent_index": 5, "subtype": "inline"}, {"bbox": [302, 341, 329, 351], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [421, 341, 466, 351], "content": "c\\in\\mathrm{Cl}_{2}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 353, 221, 363], "content": "\\mathrm{Cl}_{2}(K)", "parent_index": 5, "subtype": "inline"}, {"bbox": [337, 352, 386, 361], "content": "C^{2}=C^{1+\\sigma}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 366, 160, 373], "content": "C^{\\sigma}=C", "parent_index": 5, "subtype": "inline"}, {"bbox": [284, 366, 293, 373], "content": "K", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 377, 223, 387], "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [429, 415, 449, 425], "content": "L/\\mathbb{Q}", "parent_index": 7, "subtype": "inline"}, {"bbox": [208, 428, 216, 435], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [231, 428, 239, 437], "content": "\\mathbb{Q}", "parent_index": 7, "subtype": "inline"}, {"bbox": [326, 427, 396, 437], "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "parent_index": 7, "subtype": "inline"}, {"bbox": [209, 440, 222, 448], "content": "D_{4}", "parent_index": 7, "subtype": "inline"}, {"bbox": [320, 438, 455, 450], "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "parent_index": 7, "subtype": "inline"}, {"bbox": [245, 560, 337, 570], "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "parent_index": 9, "subtype": "inline"}, {"bbox": [360, 560, 452, 571], "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "parent_index": 9, "subtype": "inline"}, {"bbox": [316, 572, 338, 583], "content": "L/k_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [362, 572, 384, 583], "content": "L/k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [421, 573, 432, 582], "content": "E_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [456, 572, 467, 583], "content": "E_{i}^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [208, 585, 221, 594], "content": "K_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [243, 584, 255, 595], "content": "K_{i}^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [367, 587, 376, 594], "content": "\\kappa_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [230, 596, 354, 606], "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "parent_index": 9, "subtype": "inline"}, {"bbox": [372, 597, 404, 606], "content": "i=1,2", "parent_index": 9, "subtype": "inline"}, {"bbox": [377, 608, 442, 618], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 621, 154, 630], "content": "\\alpha\\in k_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [182, 619, 236, 631], "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [270, 620, 304, 628], "content": "a=\\alpha\\alpha^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [369, 623, 376, 628], "content": "\\alpha", "parent_index": 10, "subtype": "inline"}, {"bbox": [448, 619, 482, 630], "content": "\\gamma=\\sqrt{\\alpha}", "parent_index": 10, "subtype": "inline"}, {"bbox": [150, 633, 184, 642], "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "parent_index": 10, "subtype": "inline"}, {"bbox": [217, 633, 252, 642], "content": "\\gamma\\ \\in\\ K_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [299, 631, 377, 642], "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "parent_index": 10, "subtype": "inline"}, {"bbox": [471, 631, 482, 640], "content": "\\sigma^{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [142, 658, 466, 672], "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "parent_index": 11, "subtype": "interline"}, {"bbox": [192, 678, 233, 689], "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [335, 678, 374, 689], "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "parent_index": 12, "subtype": "inline"}, {"bbox": [453, 676, 486, 688], "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 690, 185, 701], "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [250, 690, 276, 701], "content": "K_{1}/\\mathbb{Q}", "parent_index": 12, "subtype": "inline"}]
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[]
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Recall that a quadratic extension $K\,=\,k({\sqrt{\alpha}}\,)$ is called essentially ramified if $\alpha{\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\alpha$ .
Proposition 6. Let $L/\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ .
Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed.
We start by remarking that if $\zeta\eta$ becomes a square in $L$ , where $\zeta$ is a root of unity in $L$ , then so does one of $\pm\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\langle\zeta_{6}\rangle$ , and here $\zeta_{6}=-\zeta_{3}^{2}$ .
Now we prove that $q_{1}\leq2$ under the assumptions we made; the claim $q_{2}\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\sqrt{\pm\varepsilon}\notin L$ . Suppose otherwise; then $k_{1}(\sqrt{\pm\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\prime}$ or $K$ . If $k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$ , then $K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$ and $K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$ . (Here and below $x^{\prime}\,=\,x^{\sigma}$ .) This however cannot occur since by assumption $\varepsilon\varepsilon^{\prime}\,=\,-1$ implying that $\sqrt{-1}\in L$ , a contradiction. Similarly, if $k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$ , then again $\sqrt{-1}\in L$ .
Thus $\sqrt{\pm\varepsilon}~\notin~L$ , and $E_{1}~=~\langle-1,\varepsilon,\eta\rangle$ for some unit $\eta\ \in\ E_{1}$ . Suppose that $\sqrt{u\eta}\in L$ for some unit $u\in k_{1}$ . Then $L=K_{1}(\sqrt{u\eta}\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\sqrt{u\eta^{\prime}}\notin L$ , hence either $E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$ and $q_{1}=1$ or $E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$ for some unit $u\in k_{1}$ and $q_{1}=2$ . Here $\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ .
Next consider the case where $k_{1}$ is complex, and let $\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\pm\varepsilon$ stays fundamental in $L$ by the argument above.
Let $\eta$ be a fundamental unit in $K_{1}$ . If $\pm\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\geq4$ , then $\pm\varepsilon\eta=\alpha^{2}$ is a square in $L$ . Applying $\tau$ to this relation we find that $-1=\varepsilon\varepsilon^{\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\sqrt{-1}$ . 口
Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}=\langle[6]\rangle$ , where $K_{2}=$ $k_{2}(\sqrt{\beta}\,)$ and $(\beta)=\mathfrak{b}^{2}$ .
Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}\neq1$ : in fact, in this case we have $(\beta)\;=\;6^{2}$ , and if we had $\,\kappa_{2}\,=\,1$ , then $\mathfrak{b}$ would have to be principal, say ${\mathfrak{b}}=(\gamma)$ . This implies that $\beta\,=\,\varepsilon\gamma^{2}$ for some unit $\varepsilon\in k_{2}$ , which in view of $q_{2}=1$ implies that $\varepsilon$ must be a square. But then $\beta$ would be a square, and this is impossible.
Conversely, suppose $\kappa_{2}\neq1$ . Let $\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\mathfrak{a}}=(\alpha)$ in $K_{2}$ . Then $\alpha^{1-\sigma^{2}}=\eta$ for some unit $\eta\in E_{2}$ , and similarly $\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$ , where $\eta^{\prime}$ is a unit in $E_{2}^{\prime}$ . But then $\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$ $N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$ in $L^{\times}$ , where $\underline{{\underline{{2}}}}$ means equal up to a square in $L^{\times}$ . Thus $\pm\eta\eta^{\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\pm\eta\eta^{\prime}$ must be a square in $k_{2}$ . The same argument show that $\pm\eta/\eta^{\prime}$ is a square in $k_{2}$ , hence we find $\eta\in k_{2}$ . Thus $\alpha^{1-\sigma^{2}}$ is fixed by $\sigma^{2}$ and so $\beta:=\alpha^{2}\in k_{2}$ . This gives $K_{2}=k_{2}(\sqrt{\beta}\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\sim{\mathfrak{b}}$ . 冏口
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<p data-bbox="125 111 486 136">Recall that a quadratic extension $K\,=\,k({\sqrt{\alpha}}\,)$ is called essentially ramified if $\alpha{\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\alpha$ . </p>
<p data-bbox="125 142 486 179">Proposition 6. Let $L/\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . </p>
<p data-bbox="125 185 486 269">Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. </p>
<p data-bbox="125 270 486 306">We start by remarking that if $\zeta\eta$ becomes a square in $L$ , where $\zeta$ is a root of unity in $L$ , then so does one of $\pm\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\langle\zeta_{6}\rangle$ , and here $\zeta_{6}=-\zeta_{3}^{2}$ . </p>
<p data-bbox="125 306 486 390">Now we prove that $q_{1}\leq2$ under the assumptions we made; the claim $q_{2}\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\sqrt{\pm\varepsilon}\notin L$ . Suppose otherwise; then $k_{1}(\sqrt{\pm\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\prime}$ or $K$ . If $k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$ , then $K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$ and $K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$ . (Here and below $x^{\prime}\,=\,x^{\sigma}$ .) This however cannot occur since by assumption $\varepsilon\varepsilon^{\prime}\,=\,-1$ implying that $\sqrt{-1}\in L$ , a contradiction. Similarly, if $k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$ , then again $\sqrt{-1}\in L$ . </p>
<p data-bbox="125 390 486 450">Thus $\sqrt{\pm\varepsilon}~\notin~L$ , and $E_{1}~=~\langle-1,\varepsilon,\eta\rangle$ for some unit $\eta\ \in\ E_{1}$ . Suppose that $\sqrt{u\eta}\in L$ for some unit $u\in k_{1}$ . Then $L=K_{1}(\sqrt{u\eta}\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\sqrt{u\eta^{\prime}}\notin L$ , hence either $E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$ and $q_{1}=1$ or $E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$ for some unit $u\in k_{1}$ and $q_{1}=2$ . Here $\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . </p>
<p data-bbox="125 451 486 474">Next consider the case where $k_{1}$ is complex, and let $\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\pm\varepsilon$ stays fundamental in $L$ by the argument above. </p>
<p data-bbox="125 475 486 522">Let $\eta$ be a fundamental unit in $K_{1}$ . If $\pm\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\geq4$ , then $\pm\varepsilon\eta=\alpha^{2}$ is a square in $L$ . Applying $\tau$ to this relation we find that $-1=\varepsilon\varepsilon^{\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\sqrt{-1}$ . 口 </p>
<p data-bbox="125 530 486 567">Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}=\langle[6]\rangle$ , where $K_{2}=$ $k_{2}(\sqrt{\beta}\,)$ and $(\beta)=\mathfrak{b}^{2}$ . </p>
<p data-bbox="125 573 486 632">Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}\neq1$ : in fact, in this case we have $(\beta)\;=\;6^{2}$ , and if we had $\,\kappa_{2}\,=\,1$ , then $\mathfrak{b}$ would have to be principal, say ${\mathfrak{b}}=(\gamma)$ . This implies that $\beta\,=\,\varepsilon\gamma^{2}$ for some unit $\varepsilon\in k_{2}$ , which in view of $q_{2}=1$ implies that $\varepsilon$ must be a square. But then $\beta$ would be a square, and this is impossible. </p>
<p data-bbox="124 633 486 700">Conversely, suppose $\kappa_{2}\neq1$ . Let $\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\mathfrak{a}}=(\alpha)$ in $K_{2}$ . Then $\alpha^{1-\sigma^{2}}=\eta$ for some unit $\eta\in E_{2}$ , and similarly $\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$ , where $\eta^{\prime}$ is a unit in $E_{2}^{\prime}$ . But then $\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$ $N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$ in $L^{\times}$ , where $\underline{{\underline{{2}}}}$ means equal up to a square in $L^{\times}$ . Thus $\pm\eta\eta^{\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\pm\eta\eta^{\prime}$ must be a square in $k_{2}$ . The same argument show that $\pm\eta/\eta^{\prime}$ is a square in $k_{2}$ , hence we find $\eta\in k_{2}$ . Thus $\alpha^{1-\sigma^{2}}$ is fixed by $\sigma^{2}$ and so $\beta:=\alpha^{2}\in k_{2}$ . This gives $K_{2}=k_{2}(\sqrt{\beta}\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\sim{\mathfrak{b}}$ . 冏口 </p>
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0003244v1
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[{"type": "text", "text": "Recall that a quadratic extension $K\\,=\\,k({\\sqrt{\\alpha}}\\,)$ is called essentially ramified if $\\alpha{\\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\\alpha$ . ", "page_idx": 7}, {"type": "text", "text": "Proposition 6. Let $L/\\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. ", "page_idx": 7}, {"type": "text", "text": "We start by remarking that if $\\zeta\\eta$ becomes a square in $L$ , where $\\zeta$ is a root of unity in $L$ , then so does one of $\\pm\\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\\langle\\zeta_{6}\\rangle$ , and here $\\zeta_{6}=-\\zeta_{3}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Now we prove that $q_{1}\\leq2$ under the assumptions we made; the claim $q_{2}\\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\\sqrt{\\pm\\varepsilon}\\notin L$ . Suppose otherwise; then $k_{1}(\\sqrt{\\pm\\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\\prime}$ or $K$ . If $k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}$ , then $K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)$ and $K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)$ . (Here and below $x^{\\prime}\\,=\\,x^{\\sigma}$ .) This however cannot occur since by assumption $\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1$ implying that $\\sqrt{-1}\\in L$ , a contradiction. Similarly, if $k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K$ , then again $\\sqrt{-1}\\in L$ . ", "page_idx": 7}, {"type": "text", "text": "Thus $\\sqrt{\\pm\\varepsilon}~\\notin~L$ , and $E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle$ for some unit $\\eta\\ \\in\\ E_{1}$ . Suppose that $\\sqrt{u\\eta}\\in L$ for some unit $u\\in k_{1}$ . Then $L=K_{1}(\\sqrt{u\\eta}\\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\\sqrt{u\\eta^{\\prime}}\\notin L$ , hence either $E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle$ and $q_{1}=1$ or $E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle$ for some unit $u\\in k_{1}$ and $q_{1}=2$ . Here $\\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . ", "page_idx": 7}, {"type": "text", "text": "Next consider the case where $k_{1}$ is complex, and let $\\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\\pm\\varepsilon$ stays fundamental in $L$ by the argument above. ", "page_idx": 7}, {"type": "text", "text": "Let $\\eta$ be a fundamental unit in $K_{1}$ . If $\\pm\\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\\geq4$ , then $\\pm\\varepsilon\\eta=\\alpha^{2}$ is a square in $L$ . Applying $\\tau$ to this relation we find that $-1=\\varepsilon\\varepsilon^{\\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\\sqrt{-1}$ . 口 ", "page_idx": 7}, {"type": "text", "text": "Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}=\\langle[6]\\rangle$ , where $K_{2}=$ $k_{2}(\\sqrt{\\beta}\\,)$ and $(\\beta)=\\mathfrak{b}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}\\neq1$ : in fact, in this case we have $(\\beta)\\;=\\;6^{2}$ , and if we had $\\,\\kappa_{2}\\,=\\,1$ , then $\\mathfrak{b}$ would have to be principal, say ${\\mathfrak{b}}=(\\gamma)$ . This implies that $\\beta\\,=\\,\\varepsilon\\gamma^{2}$ for some unit $\\varepsilon\\in k_{2}$ , which in view of $q_{2}=1$ implies that $\\varepsilon$ must be a square. But then $\\beta$ would be a square, and this is impossible. ", "page_idx": 7}, {"type": "text", "text": "Conversely, suppose $\\kappa_{2}\\neq1$ . Let $\\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\\mathfrak{a}}=(\\alpha)$ in $K_{2}$ . Then $\\alpha^{1-\\sigma^{2}}=\\eta$ for some unit $\\eta\\in E_{2}$ , and similarly $\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}$ , where $\\eta^{\\prime}$ is a unit in $E_{2}^{\\prime}$ . But then $\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}$ $N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1$ in $L^{\\times}$ , where $\\underline{{\\underline{{2}}}}$ means equal up to a square in $L^{\\times}$ . Thus $\\pm\\eta\\eta^{\\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\\pm\\eta\\eta^{\\prime}$ must be a square in $k_{2}$ . The same argument show that $\\pm\\eta/\\eta^{\\prime}$ is a square in $k_{2}$ , hence we find $\\eta\\in k_{2}$ . Thus $\\alpha^{1-\\sigma^{2}}$ is fixed by $\\sigma^{2}$ and so $\\beta:=\\alpha^{2}\\in k_{2}$ . This gives $K_{2}=k_{2}(\\sqrt{\\beta}\\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\\sim{\\mathfrak{b}}$ . 冏口 ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "口", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}], "index": 43}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 101], "spans": [{"bbox": [126, 93, 131, 101], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 114, 486, 138]}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [126, 145, 487, 182]}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8, "bbox_fs": [125, 188, 487, 272]}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "bbox_fs": [125, 271, 486, 307]}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18, "bbox_fs": [125, 307, 487, 392]}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24, "bbox_fs": [126, 390, 486, 452]}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [126, 451, 486, 476]}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "口", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [125, 477, 487, 524]}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "bbox_fs": [126, 532, 487, 568]}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38, "bbox_fs": [124, 574, 487, 635]}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}, {"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text", "cross_page": true}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text", "cross_page": true}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26, "cross_page": true}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text", "cross_page": true}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text", "cross_page": true}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25, "cross_page": true}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text", "cross_page": true}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11, "cross_page": true}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text", "cross_page": true}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58, "cross_page": true}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60, "cross_page": true}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text", "cross_page": true}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text", "cross_page": true}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23, "cross_page": true}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text", "cross_page": true}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "冏口", "type": "text", "cross_page": true}], "index": 2}], "index": 43, "bbox_fs": [123, 635, 487, 702]}]}
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[{"type": "text", "bbox": [125, 111, 486, 136], "content": "Recall that a quadratic extension is called essentially ramified if is not an ideal square. This definition is independent of the choice of .", "index": 0}, {"type": "text", "bbox": [125, 142, 486, 179], "content": "Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then .", "index": 1}, {"type": "text", "bbox": [125, 185, 486, 269], "content": "Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed.", "index": 2}, {"type": "text", "bbox": [125, 270, 486, 306], "content": "We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here .", "index": 3}, {"type": "text", "bbox": [125, 306, 486, 390], "content": "Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again .", "index": 4}, {"type": "text", "bbox": [125, 390, 486, 450], "content": "Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of .", "index": 5}, {"type": "text", "bbox": [125, 451, 486, 474], "content": "Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above.", "index": 6}, {"type": "text", "bbox": [125, 475, 486, 522], "content": "Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口", "index": 7}, {"type": "text", "bbox": [125, 530, 486, 567], "content": "Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and .", "index": 8}, {"type": "text", "bbox": [125, 573, 486, 632], "content": "Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible.", "index": 9}, {"type": "text", "bbox": [124, 633, 486, 700], "content": "Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . This gives , hence is not essentially ramified, and moreover, . 冏口", "index": 10}]
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[{"bbox": [137, 114, 486, 126], "content": "Recall that a quadratic extension is called essentially ramified if", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 462, 138], "content": "is not an ideal square. This definition is independent of the choice of .", "parent_index": 0, "line_index": 1}, {"bbox": [126, 145, 487, 158], "content": "Proposition 6. Let be a non-CM totally complex dihedral extension not con-", "parent_index": 1, "line_index": 0}, {"bbox": [126, 157, 487, 170], "content": "taining , and assume that and are essentially ramified. If the", "parent_index": 1, "line_index": 1}, {"bbox": [126, 169, 481, 182], "content": "fundamental unit of the real quadratic subfield of has norm , then .", "parent_index": 1, "line_index": 2}, {"bbox": [127, 188, 485, 200], "content": "Proof. Notice first that cannot be real (in fact, is not totally real by assumption,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 199, 486, 212], "content": "and since is a cyclic quartic extension, no infinite prime can ramify in );", "parent_index": 2, "line_index": 1}, {"bbox": [126, 212, 486, 224], "content": "thus exactly one of , is real, and the other is complex. Multiplying the class", "parent_index": 2, "line_index": 2}, {"bbox": [125, 224, 486, 236], "content": "number formulas, Proposition 3, for and (note that since both", "parent_index": 2, "line_index": 3}, {"bbox": [126, 236, 487, 248], "content": "and are essentially ramified) we find that is a square. If we can", "parent_index": 2, "line_index": 4}, {"bbox": [125, 248, 486, 260], "content": "prove that , then is a square between 2 and 8, which implies that", "parent_index": 2, "line_index": 5}, {"bbox": [125, 259, 340, 272], "content": "we must have and as claimed.", "parent_index": 2, "line_index": 6}, {"bbox": [137, 271, 485, 284], "content": "We start by remarking that if becomes a square in , where is a root of unity", "parent_index": 3, "line_index": 0}, {"bbox": [126, 284, 486, 295], "content": "in , then so does one of . This follows from the fact that the only non-trivial", "parent_index": 3, "line_index": 1}, {"bbox": [125, 295, 485, 307], "content": "roots of unity that can be in are the sixth roots of unity , and here .", "parent_index": 3, "line_index": 2}, {"bbox": [137, 307, 486, 319], "content": "Now we prove that under the assumptions we made; the claim will", "parent_index": 4, "line_index": 0}, {"bbox": [125, 320, 487, 330], "content": "then follow by symmetry. Assume first that is real and let be the fundamental", "parent_index": 4, "line_index": 1}, {"bbox": [126, 331, 487, 344], "content": "unit of . We claim that . Suppose otherwise; then is one of", "parent_index": 4, "line_index": 2}, {"bbox": [126, 343, 487, 356], "content": ", or . If , then and . (Here", "parent_index": 4, "line_index": 3}, {"bbox": [125, 355, 485, 369], "content": "and below .) This however cannot occur since by assumption", "parent_index": 4, "line_index": 4}, {"bbox": [125, 368, 487, 380], "content": "implying that , a contradiction. Similarly, if , then again", "parent_index": 4, "line_index": 5}, {"bbox": [126, 380, 170, 392], "content": ".", "parent_index": 4, "line_index": 6}, {"bbox": [137, 390, 486, 405], "content": "Thus , and for some unit . Suppose that", "parent_index": 5, "line_index": 0}, {"bbox": [126, 404, 485, 417], "content": "for some unit . Then , contradicting our assumption", "parent_index": 5, "line_index": 1}, {"bbox": [126, 417, 486, 429], "content": "that is essentially ramified. The same argument shows that , hence", "parent_index": 5, "line_index": 2}, {"bbox": [126, 428, 484, 442], "content": "either and or for some unit", "parent_index": 5, "line_index": 3}, {"bbox": [126, 441, 479, 452], "content": "and . Here is a root of unity generating the torsion subgroup of .", "parent_index": 5, "line_index": 4}, {"bbox": [137, 451, 486, 465], "content": "Next consider the case where is complex, and let denote the fundamental", "parent_index": 6, "line_index": 0}, {"bbox": [126, 464, 425, 476], "content": "unit of . Then stays fundamental in by the argument above.", "parent_index": 6, "line_index": 1}, {"bbox": [137, 477, 485, 488], "content": "Let be a fundamental unit in . If became a square in , then clearly", "parent_index": 7, "line_index": 0}, {"bbox": [126, 488, 487, 501], "content": "could not be essentially ramified. Thus if we have , then is", "parent_index": 7, "line_index": 1}, {"bbox": [125, 500, 487, 513], "content": "a square in . Applying to this relation we find that is a square in ,", "parent_index": 7, "line_index": 2}, {"bbox": [125, 512, 486, 524], "content": "contradicting the assumption that does not contain . 口", "parent_index": 7, "line_index": 3}, {"bbox": [126, 532, 487, 545], "content": "Proposition 7. Suppose that . Then is essentially ramified if and", "parent_index": 8, "line_index": 0}, {"bbox": [127, 543, 486, 558], "content": "only if ; if is not essentially ramified, then , where", "parent_index": 8, "line_index": 1}, {"bbox": [126, 556, 223, 568], "content": "and .", "parent_index": 8, "line_index": 2}, {"bbox": [126, 574, 485, 587], "content": "Proof. First notice that if is not essentially ramified, then : in fact,", "parent_index": 9, "line_index": 0}, {"bbox": [124, 586, 486, 599], "content": "in this case we have , and if we had , then would have to be", "parent_index": 9, "line_index": 1}, {"bbox": [126, 599, 486, 611], "content": "principal, say . This implies that for some unit , which in", "parent_index": 9, "line_index": 2}, {"bbox": [125, 611, 487, 623], "content": "view of implies that must be a square. But then would be a square, and", "parent_index": 9, "line_index": 3}, {"bbox": [125, 623, 204, 635], "content": "this is impossible.", "parent_index": 9, "line_index": 4}, {"bbox": [138, 635, 487, 647], "content": "Conversely, suppose . Let be a nonprincipal ideal in of absolute norm", "parent_index": 10, "line_index": 0}, {"bbox": [126, 646, 486, 660], "content": ", and assume that in . Then for some unit , and", "parent_index": 10, "line_index": 1}, {"bbox": [123, 658, 485, 676], "content": "similarly , where is a unit in . But then", "parent_index": 10, "line_index": 2}, {"bbox": [125, 674, 487, 690], "content": "in , where means equal up to a square in", "parent_index": 10, "line_index": 3}, {"bbox": [126, 689, 484, 702], "content": ". Thus is a square in , so our assumption that implies that", "parent_index": 10, "line_index": 4}, {"bbox": [124, 114, 486, 127], "content": "must be a square in . The same argument show that is a square in ,", "parent_index": 10, "line_index": 5}, {"bbox": [124, 126, 487, 140], "content": "hence we find . Thus is fixed by and so . This gives", "parent_index": 10, "line_index": 6}, {"bbox": [126, 139, 487, 151], "content": ", hence is not essentially ramified, and moreover, . 冏口", "parent_index": 10, "line_index": 7}]
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[{"bbox": [291, 115, 346, 126], "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 145, 137], "content": "\\alpha{\\mathcal{O}}_{k}", "parent_index": 0, "subtype": "inline"}, {"bbox": [452, 131, 459, 135], "content": "\\alpha", "parent_index": 0, "subtype": "inline"}, {"bbox": [218, 146, 239, 156], "content": "L/\\mathbb{Q}", "parent_index": 1, "subtype": "inline"}, {"bbox": [161, 157, 182, 168], "content": "\\sqrt{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [268, 158, 293, 169], "content": "L/K_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [317, 158, 342, 169], "content": "L/K_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [342, 171, 351, 178], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [399, 171, 412, 179], "content": "^{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [440, 171, 477, 180], "content": "q_{1}q_{2}=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [227, 189, 233, 197], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [334, 190, 343, 197], "content": "K", "parent_index": 2, "subtype": "inline"}, {"bbox": [170, 201, 188, 211], "content": "L/k", "parent_index": 2, "subtype": "inline"}, {"bbox": [459, 201, 478, 211], "content": "K/k", "parent_index": 2, "subtype": "inline"}, {"bbox": [213, 213, 223, 222], "content": "k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [229, 213, 239, 222], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [289, 225, 311, 235], "content": "L/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [334, 225, 356, 235], "content": "L/k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [409, 226, 435, 233], "content": "\\upsilon=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 237, 151, 247], "content": "L/K_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [173, 237, 198, 247], "content": "L/K_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [365, 238, 388, 247], "content": "2q_{1}q_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [175, 250, 217, 259], "content": "q_{1},q_{2}\\leq2", "parent_index": 2, "subtype": "inline"}, {"bbox": [247, 250, 270, 259], "content": "2q_{1}q_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [189, 262, 230, 271], "content": "2q_{1}q_{2}=4", "parent_index": 2, "subtype": "inline"}, {"bbox": [252, 262, 288, 270], "content": "q_{1}q_{2}=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [266, 273, 276, 282], "content": "\\zeta\\eta", "parent_index": 3, "subtype": "inline"}, {"bbox": [366, 273, 374, 280], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [407, 273, 412, 282], "content": "\\zeta", "parent_index": 3, "subtype": "inline"}, {"bbox": [138, 285, 145, 293], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [239, 286, 253, 294], "content": "\\pm\\eta", "parent_index": 3, "subtype": "inline"}, {"bbox": [251, 297, 258, 304], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [380, 297, 397, 307], "content": "\\langle\\zeta_{6}\\rangle", "parent_index": 3, "subtype": "inline"}, {"bbox": [442, 296, 482, 307], "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [222, 309, 249, 318], "content": "q_{1}\\leq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [439, 309, 466, 318], "content": "q_{2}\\leq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [318, 321, 327, 330], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [393, 324, 397, 328], "content": "\\varepsilon", "parent_index": 4, "subtype": "inline"}, {"bbox": [159, 333, 169, 342], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [244, 331, 285, 342], "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "parent_index": 4, "subtype": "inline"}, {"bbox": [404, 331, 444, 343], "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 346, 139, 354], "content": "K_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 345, 158, 356], "content": "K_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 346, 183, 353], "content": "K", "parent_index": 4, "subtype": "inline"}, {"bbox": [201, 344, 268, 356], "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [297, 344, 367, 356], "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [390, 344, 453, 356], "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "parent_index": 4, "subtype": "inline"}, {"bbox": [175, 357, 212, 365], "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "parent_index": 4, "subtype": "inline"}, {"bbox": [443, 357, 485, 366], "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "parent_index": 4, "subtype": "inline"}, {"bbox": [190, 368, 232, 379], "content": "\\sqrt{-1}\\in L", "parent_index": 4, "subtype": "inline"}, {"bbox": [367, 369, 432, 380], "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 380, 166, 391], "content": "\\sqrt{-1}\\in L", "parent_index": 4, "subtype": "inline"}, {"bbox": [164, 392, 210, 403], "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "parent_index": 5, "subtype": "inline"}, {"bbox": [239, 393, 308, 403], "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [379, 394, 414, 403], "content": "\\eta\\ \\in\\ E_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 406, 164, 416], "content": "\\sqrt{u\\eta}\\in L", "parent_index": 5, "subtype": "inline"}, {"bbox": [228, 406, 256, 415], "content": "u\\in k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [289, 405, 351, 416], "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [147, 418, 172, 428], "content": "L/K_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [414, 417, 455, 428], "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "parent_index": 5, "subtype": "inline"}, {"bbox": [155, 430, 227, 440], "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [251, 431, 279, 439], "content": "q_{1}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [296, 429, 389, 440], "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [455, 430, 484, 439], "content": "u\\in k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [145, 443, 173, 451], "content": "q_{1}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [203, 442, 208, 451], "content": "\\zeta", "parent_index": 5, "subtype": "inline"}, {"bbox": [431, 442, 446, 451], "content": "W_{L}", "parent_index": 5, "subtype": "inline"}, {"bbox": [461, 442, 475, 451], "content": "E_{L}", "parent_index": 5, "subtype": "inline"}, {"bbox": [271, 454, 280, 463], "content": "k_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [371, 457, 376, 461], "content": "\\varepsilon", "parent_index": 6, "subtype": "inline"}, {"bbox": [158, 466, 168, 475], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [201, 466, 214, 474], "content": "\\pm\\varepsilon", "parent_index": 6, "subtype": "inline"}, {"bbox": [311, 466, 318, 473], "content": "L", "parent_index": 6, "subtype": "inline"}, {"bbox": [156, 481, 162, 488], "content": "\\eta", "parent_index": 7, "subtype": "inline"}, {"bbox": [280, 478, 294, 487], "content": "K_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [313, 479, 326, 488], "content": "\\pm\\eta", "parent_index": 7, "subtype": "inline"}, {"bbox": [419, 478, 426, 486], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 489, 151, 500], "content": "L/K_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [376, 491, 403, 499], "content": "q_{1}\\geq4", "parent_index": 7, "subtype": "inline"}, {"bbox": [432, 489, 474, 500], "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [178, 502, 185, 509], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [236, 505, 241, 509], "content": "\\tau", "parent_index": 7, "subtype": "inline"}, {"bbox": [371, 501, 409, 510], "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [475, 502, 482, 509], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [278, 514, 285, 521], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [363, 513, 385, 523], "content": "\\sqrt{-1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [261, 534, 291, 543], "content": "q_{2}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [327, 533, 355, 544], "content": "K_{2}/k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [158, 546, 189, 555], "content": "\\kappa_{2}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [206, 545, 234, 556], "content": "K_{2}/k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [380, 545, 424, 556], "content": "\\kappa_{2}=\\langle[6]\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [460, 545, 486, 555], "content": "K_{2}=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 556, 160, 568], "content": "k_{2}(\\sqrt{\\beta}\\,)", "parent_index": 8, "subtype": "inline"}, {"bbox": [182, 556, 219, 568], "content": "(\\beta)=\\mathfrak{b}^{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [243, 576, 271, 586], "content": "K_{2}/k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [416, 577, 446, 586], "content": "\\kappa_{2}\\neq1", "parent_index": 9, "subtype": "inline"}, {"bbox": [221, 587, 261, 598], "content": "(\\beta)\\;=\\;6^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [335, 589, 367, 597], "content": "\\,\\kappa_{2}\\,=\\,1", "parent_index": 9, "subtype": "inline"}, {"bbox": [398, 588, 403, 596], "content": "\\mathfrak{b}", "parent_index": 9, "subtype": "inline"}, {"bbox": [189, 600, 222, 610], "content": "{\\mathfrak{b}}=(\\gamma)", "parent_index": 9, "subtype": "inline"}, {"bbox": [310, 600, 346, 610], "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [412, 601, 441, 609], "content": "\\varepsilon\\in k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [160, 613, 187, 622], "content": "q_{2}=1", "parent_index": 9, "subtype": "inline"}, {"bbox": [245, 615, 249, 620], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [375, 613, 381, 622], "content": "\\beta", "parent_index": 9, "subtype": "inline"}, {"bbox": [227, 636, 255, 646], "content": "\\kappa_{2}\\neq1", "parent_index": 10, "subtype": "inline"}, {"bbox": [280, 639, 285, 644], "content": "\\mathfrak{a}", "parent_index": 10, "subtype": "inline"}, {"bbox": [400, 636, 409, 645], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 652, 132, 657], "content": "a", "parent_index": 10, "subtype": "inline"}, {"bbox": [216, 649, 250, 659], "content": "{\\mathfrak{a}}=(\\alpha)", "parent_index": 10, "subtype": "inline"}, {"bbox": [267, 649, 280, 658], "content": "K_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [316, 646, 363, 659], "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "parent_index": 10, "subtype": "inline"}, {"bbox": [430, 649, 462, 659], "content": "\\eta\\in E_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [168, 661, 219, 673], "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [256, 663, 264, 673], "content": "\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [324, 663, 336, 673], "content": "E_{2}^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [390, 660, 485, 673], "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "parent_index": 10, "subtype": "inline"}, {"bbox": [125, 675, 269, 689], "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "parent_index": 10, "subtype": "inline"}, {"bbox": [285, 677, 299, 686], "content": "L^{\\times}", "parent_index": 10, "subtype": "inline"}, {"bbox": [336, 675, 344, 686], "content": "\\underline{{\\underline{{2}}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 690, 140, 698], "content": "L^{\\times}", "parent_index": 10, "subtype": "inline"}, {"bbox": [172, 690, 194, 700], "content": "\\pm\\eta\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [259, 691, 266, 698], "content": "L", "parent_index": 10, "subtype": "inline"}, {"bbox": [376, 691, 405, 700], "content": "q_{2}=1", "parent_index": 10, "subtype": "inline"}, {"bbox": [464, 690, 484, 700], "content": "\\pm\\eta\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [219, 116, 229, 125], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [378, 115, 404, 126], "content": "\\pm\\eta/\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [472, 116, 482, 125], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [189, 129, 217, 139], "content": "\\eta\\in k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [251, 126, 276, 137], "content": "\\alpha^{1-\\sigma^{2}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [329, 128, 340, 137], "content": "\\sigma^{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [375, 128, 433, 139], "content": "\\beta:=\\alpha^{2}\\in k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 140, 186, 151], "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [219, 141, 247, 151], "content": "K_{2}/k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [433, 141, 456, 149], "content": "a\\sim{\\mathfrak{b}}", "parent_index": 10, "subtype": "inline"}]
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From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\mathrm{B}$ ) as explained in the Introduction. Let
Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\,=\,\mathbb{Q}({\sqrt{d}}\,)$ are constructed by choosing a “primitive” solution $\left(x,y,z\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$ and putting $L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$ with $\alpha=z+x\sqrt{d_{1}d_{2}}$ (primitive here means that $\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$ . If we put $\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$ , then it is an elementary exerc i se to show that $\alpha\beta$ is a square in $L$ , hence we also have $L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\mathfrak{p}}_{3}^{u}=(\pi_{3})$ is principal. Again it can be checked that $\beta=\pm\pi_{3}$ for a suitable choice of the sign.
Example. Consider the case $d\,=\,-31\cdot5\cdot8$ ; here $\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$ , and the positive sign is correct since $3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$ mod 4 is primary. The minimal polynomial of $\sqrt{\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1.
The fields $K_{2}=k_{2}(\sqrt{\alpha}\,)$ and $\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\sqrt{\alpha}}\,)$ for $F=k_{2}(\sqrt{d_{2}}\,)$ , and it is the ambiguous class group $\mathrm{Am}(M/F)$ that contains the information we are interested in.
Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\#\operatorname{Am}(M/F)\mid$ 2. In particular, $\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial).
Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd.
Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\,=\,F({\sqrt{\alpha}}\,)$ with $\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\alpha$ to an odd power, so $t\ =\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$ , and $\mathrm{Cl_{2}}(M)$ is cyclic.
In case B), however, $F$ is real; since $\alpha\,\in\,k_{2}$ has norm $d_{3}y^{2}\,<\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\#\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\mathbb{Z}$ -rank 2, i.e. $(E:H)\geq4$ by consideration of the infinite primes alone. In particular, $\#\operatorname{Am}_{2}(M/F)\leq2$ in case B). 口
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<p data-bbox="125 157 486 181">From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\mathrm{B}$ ) as explained in the Introduction. Let </p>
<p data-bbox="124 214 486 385">Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\,=\,\mathbb{Q}({\sqrt{d}}\,)$ are constructed by choosing a “primitive” solution $\left(x,y,z\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$ and putting $L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$ with $\alpha=z+x\sqrt{d_{1}d_{2}}$ (primitive here means that $\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$ . If we put $\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$ , then it is an elementary exerc i se to show that $\alpha\beta$ is a square in $L$ , hence we also have $L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\mathfrak{p}}_{3}^{u}=(\pi_{3})$ is principal. Again it can be checked that $\beta=\pm\pi_{3}$ for a suitable choice of the sign. </p>
<p data-bbox="125 389 486 427">Example. Consider the case $d\,=\,-31\cdot5\cdot8$ ; here $\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$ , and the positive sign is correct since $3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$ mod 4 is primary. The minimal polynomial of $\sqrt{\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. </p>
<p data-bbox="125 433 486 483">The fields $K_{2}=k_{2}(\sqrt{\alpha}\,)$ and $\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\sqrt{\alpha}}\,)$ for $F=k_{2}(\sqrt{d_{2}}\,)$ , and it is the ambiguous class group $\mathrm{Am}(M/F)$ that contains the information we are interested in. </p>
<p data-bbox="125 489 486 513">Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\#\operatorname{Am}(M/F)\mid$ 2. In particular, $\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). </p>
<p data-bbox="126 519 486 567">Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. </p>
<p data-bbox="125 568 486 640">Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\,=\,F({\sqrt{\alpha}}\,)$ with $\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\alpha$ to an odd power, so $t\ =\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$ , and $\mathrm{Cl_{2}}(M)$ is cyclic. </p>
<p data-bbox="125 640 486 700">In case B), however, $F$ is real; since $\alpha\,\in\,k_{2}$ has norm $d_{3}y^{2}\,<\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\#\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\mathbb{Z}$ -rank 2, i.e. $(E:H)\geq4$ by consideration of the infinite primes alone. In particular, $\#\operatorname{Am}_{2}(M/F)\leq2$ in case B). 口 </p>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\\mathrm{B}$ ) as explained in the Introduction. Let ", "page_idx": 8}, {"type": "text", "text": "Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)$ are constructed by choosing a “primitive” solution $\\left(x,y,z\\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}$ and putting $L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)$ with $\\alpha=z+x\\sqrt{d_{1}d_{2}}$ (primitive here means that $\\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)$ . If we put $\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)$ , then it is an elementary exerc i se to show that $\\alpha\\beta$ is a square in $L$ , hence we also have $L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\\mathfrak{p}}_{3}^{u}=(\\pi_{3})$ is principal. Again it can be checked that $\\beta=\\pm\\pi_{3}$ for a suitable choice of the sign. ", "page_idx": 8}, {"type": "text", "text": "Example. Consider the case $d\\,=\\,-31\\cdot5\\cdot8$ ; here $\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)$ , and the positive sign is correct since $3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}$ mod 4 is primary. The minimal polynomial of $\\sqrt{\\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. ", "page_idx": 8}, {"type": "text", "text": "The fields $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ and $\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\\sqrt{\\alpha}}\\,)$ for $F=k_{2}(\\sqrt{d_{2}}\\,)$ , and it is the ambiguous class group $\\mathrm{Am}(M/F)$ that contains the information we are interested in. ", "page_idx": 8}, {"type": "text", "text": "Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\\#\\operatorname{Am}(M/F)\\mid$ 2. In particular, $\\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). ", "page_idx": 8}, {"type": "text", "text": "Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. ", "page_idx": 8}, {"type": "text", "text": "Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\\,=\\,F({\\sqrt{\\alpha}}\\,)$ with $\\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\\alpha$ to an odd power, so $t\\ =\\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2$ , and $\\mathrm{Cl_{2}}(M)$ is cyclic. ", "page_idx": 8}, {"type": "text", "text": "In case B), however, $F$ is real; since $\\alpha\\,\\in\\,k_{2}$ has norm $d_{3}y^{2}\\,<\\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\\mathbb{Z}$ -rank 2, i.e. $(E:H)\\geq4$ by consideration of the infinite primes alone. In particular, $\\#\\operatorname{Am}_{2}(M/F)\\leq2$ in case B). 口 ", "page_idx": 8}]
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"poly": [894.0, 1397.0, 1303.0, 1397.0, 1303.0, 1432.0, 894.0, 1432.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [665.0, 259.0, 1035.0, 259.0, 1035.0, 280.0, 665.0, 280.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1336.0, 261.0, 1350.0, 261.0, 1350.0, 282.0, 1336.0, 282.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 8, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [{"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text"}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text"}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text"}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text"}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text"}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text"}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text"}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "冏口", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a “primitive” solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R´edei’s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [], "index": 1, "bbox_fs": [124, 114, 487, 151], "lines_deleted": true}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [126, 159, 484, 183]}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a “primitive” solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5, "bbox_fs": [124, 217, 487, 388]}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20, "bbox_fs": [126, 392, 486, 429]}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5, "bbox_fs": [125, 435, 487, 484]}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [124, 491, 487, 515]}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R´edei’s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [125, 521, 486, 570]}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5, "bbox_fs": [125, 569, 487, 641]}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}, {"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text", "cross_page": true}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7, "cross_page": true}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text", "cross_page": true}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52, "cross_page": true}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text", "cross_page": true}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80, "cross_page": true}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text", "cross_page": true}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "口", "type": "text", "cross_page": true}], "index": 1}], "index": 40, "bbox_fs": [125, 641, 486, 702]}]}
|
[{"type": "text", "bbox": [125, 112, 486, 150], "content": "", "index": 0}, {"type": "text", "bbox": [125, 157, 486, 181], "content": "From now on assume that is one of the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let", "index": 1}, {"type": "text", "bbox": [124, 214, 486, 385], "content": "Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign.", "index": 2}, {"type": "text", "bbox": [125, 389, 486, 427], "content": "Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1.", "index": 3}, {"type": "text", "bbox": [125, 433, 486, 483], "content": "The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in.", "index": 4}, {"type": "text", "bbox": [125, 489, 486, 513], "content": "Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial).", "index": 5}, {"type": "text", "bbox": [126, 519, 486, 567], "content": "Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd.", "index": 6}, {"type": "text", "bbox": [125, 568, 486, 640], "content": "Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic.", "index": 7}, {"type": "text", "bbox": [125, 640, 486, 700], "content": "In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口", "index": 8}]
|
[{"bbox": [137, 159, 484, 172], "content": "From now on assume that is one of the imaginary quadratic fields of type A)", "parent_index": 1, "line_index": 0}, {"bbox": [126, 172, 316, 183], "content": "or ) as explained in the Introduction. Let", "parent_index": 1, "line_index": 1}, {"bbox": [138, 217, 484, 228], "content": "Then there exist two unramified cyclic quartic extensions of which are", "parent_index": 2, "line_index": 0}, {"bbox": [126, 229, 486, 240], "content": "over (see Proposition 2). Let us say a few words about their construction.", "parent_index": 2, "line_index": 1}, {"bbox": [126, 240, 484, 252], "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization", "parent_index": 2, "line_index": 2}, {"bbox": [125, 252, 486, 264], "content": "implies that unramified cyclic quartic extensions of are constructed", "parent_index": 2, "line_index": 3}, {"bbox": [124, 263, 487, 278], "content": "by choosing a “primitive” solution of and putting", "parent_index": 2, "line_index": 4}, {"bbox": [126, 276, 487, 289], "content": "with (primitive here means that should not", "parent_index": 2, "line_index": 5}, {"bbox": [125, 288, 487, 301], "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "parent_index": 2, "line_index": 6}, {"bbox": [126, 300, 486, 315], "content": "then . If we put , then it is an elementary", "parent_index": 2, "line_index": 7}, {"bbox": [126, 315, 486, 326], "content": "exerc i se to show that is a square in , hence we also have etc.", "parent_index": 2, "line_index": 8}, {"bbox": [125, 326, 487, 339], "content": "If , then it is easy to see that we may choose as the fundamental unit of", "parent_index": 2, "line_index": 9}, {"bbox": [126, 338, 487, 351], "content": "; if , then genus theory says that a) the class number of is twice an", "parent_index": 2, "line_index": 10}, {"bbox": [126, 352, 486, 363], "content": "odd number ; and b) the prime ideal above in is in the principal genus, so", "parent_index": 2, "line_index": 11}, {"bbox": [126, 363, 486, 375], "content": "is principal. Again it can be checked that for a suitable choice", "parent_index": 2, "line_index": 12}, {"bbox": [125, 374, 175, 388], "content": "of the sign.", "parent_index": 2, "line_index": 13}, {"bbox": [126, 392, 486, 404], "content": "Example. Consider the case ; here , and the", "parent_index": 3, "line_index": 0}, {"bbox": [126, 404, 486, 417], "content": "positive sign is correct since mod 4 is primary. The minimal", "parent_index": 3, "line_index": 1}, {"bbox": [126, 416, 397, 429], "content": "polynomial of is : compare Table 1.", "parent_index": 3, "line_index": 2}, {"bbox": [137, 435, 486, 449], "content": "The fields and will play a dominant role in the", "parent_index": 4, "line_index": 0}, {"bbox": [125, 448, 486, 460], "content": "proof below; they are both contai n ed in for , and it is the", "parent_index": 4, "line_index": 1}, {"bbox": [126, 460, 487, 474], "content": "ambiguous class group that contains the information we are interested", "parent_index": 4, "line_index": 2}, {"bbox": [126, 473, 138, 484], "content": "in.", "parent_index": 4, "line_index": 3}, {"bbox": [124, 491, 487, 503], "content": "Lemma 6. The field has odd class number (even in the strict sense), and we", "parent_index": 5, "line_index": 0}, {"bbox": [125, 502, 469, 515], "content": "have 2. In particular, is cyclic (though possibly trivial).", "parent_index": 5, "line_index": 1}, {"bbox": [125, 521, 486, 535], "content": "Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory", "parent_index": 6, "line_index": 0}, {"bbox": [126, 533, 486, 547], "content": "[12] (since in case A) and in case", "parent_index": 6, "line_index": 1}, {"bbox": [125, 546, 486, 558], "content": "B)). Since is the Hilbert class field of in the strict sense, its class number in", "parent_index": 6, "line_index": 2}, {"bbox": [125, 558, 225, 570], "content": "the strict sense is odd.", "parent_index": 6, "line_index": 3}, {"bbox": [137, 569, 484, 581], "content": "Next we apply the ambiguous class number formula. In case A), is complex,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 581, 485, 594], "content": "and exactly the two primes above ramify in . Note that", "parent_index": 7, "line_index": 1}, {"bbox": [126, 594, 486, 606], "content": "with primary of norm ; there are four primes above in , and exactly", "parent_index": 7, "line_index": 2}, {"bbox": [126, 606, 487, 618], "content": "two of them divide to an odd power, so by the decomposition law in", "parent_index": 7, "line_index": 3}, {"bbox": [125, 617, 487, 630], "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "parent_index": 7, "line_index": 4}, {"bbox": [125, 629, 360, 641], "content": ", and is cyclic.", "parent_index": 7, "line_index": 5}, {"bbox": [137, 641, 486, 653], "content": "In case B), however, is real; since has norm , it has mixed", "parent_index": 8, "line_index": 0}, {"bbox": [125, 653, 486, 665], "content": "signature, hence there are exactly two infinite primes that ramify in . As", "parent_index": 8, "line_index": 1}, {"bbox": [125, 664, 486, 678], "content": "in case A), there are two finite primes above that ramify in , so we get", "parent_index": 8, "line_index": 2}, {"bbox": [126, 678, 485, 689], "content": ". Since has odd class number in the strict sense,", "parent_index": 8, "line_index": 3}, {"bbox": [126, 690, 486, 702], "content": "has units of independent signs. This implies that the group of units that are positive", "parent_index": 8, "line_index": 4}, {"bbox": [125, 114, 486, 126], "content": "at the two ramified infinite primes has -rank 2, i.e. by consideration", "parent_index": 8, "line_index": 5}, {"bbox": [125, 127, 486, 138], "content": "of the infinite primes alone. In particular, in case B). 口", "parent_index": 8, "line_index": 6}]
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[]
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[{"bbox": [255, 162, 261, 169], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 173, 145, 181], "content": "\\mathrm{B}", "parent_index": 1, "subtype": "inline"}, {"bbox": [414, 218, 420, 225], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [472, 218, 484, 227], "content": "D_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [149, 230, 157, 239], "content": "\\mathbb{Q}", "parent_index": 2, "subtype": "inline"}, {"bbox": [361, 242, 372, 250], "content": "C_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [433, 242, 484, 250], "content": "d=d_{1}d_{2}\\cdot d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [359, 252, 412, 264], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [283, 266, 316, 276], "content": "\\left(x,y,z\\right)", "parent_index": 2, "subtype": "inline"}, {"bbox": [332, 265, 428, 275], "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 277, 208, 288], "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 277, 305, 288], "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [429, 281, 435, 286], "content": "\\alpha", "parent_index": 2, "subtype": "inline"}, {"bbox": [149, 301, 241, 313], "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [294, 302, 372, 314], "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [221, 316, 233, 326], "content": "\\alpha\\beta", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 317, 304, 324], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [394, 315, 466, 326], "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [136, 329, 172, 337], "content": "d_{3}=-4", "parent_index": 2, "subtype": "inline"}, {"bbox": [360, 329, 366, 338], "content": "\\beta", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 340, 136, 349], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [151, 340, 187, 349], "content": "d_{3}\\neq-4", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 340, 411, 348], "content": "h", "parent_index": 2, "subtype": "inline"}, {"bbox": [425, 340, 435, 349], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [181, 355, 187, 360], "content": "u", "parent_index": 2, "subtype": "inline"}, {"bbox": [291, 354, 301, 362], "content": "{\\mathfrak{p}}_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [331, 353, 341, 361], "content": "d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [355, 353, 365, 361], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 364, 168, 374], "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "parent_index": 2, "subtype": "inline"}, {"bbox": [356, 364, 394, 374], "content": "\\beta=\\pm\\pi_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [262, 394, 329, 402], "content": "d\\,=\\,-31\\cdot5\\cdot8", "parent_index": 3, "subtype": "inline"}, {"bbox": [358, 392, 443, 404], "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "parent_index": 3, "subtype": "inline"}, {"bbox": [248, 405, 346, 416], "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [189, 418, 208, 429], "content": "\\sqrt{\\pi_{3}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [221, 417, 313, 428], "content": "f(x)=x^{4}-6x^{2}-31", "parent_index": 3, "subtype": "inline"}, {"bbox": [185, 437, 246, 448], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [269, 435, 340, 448], "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "parent_index": 4, "subtype": "inline"}, {"bbox": [297, 449, 353, 460], "content": "M=F({\\sqrt{\\alpha}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [370, 448, 428, 460], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 461, 275, 472], "content": "\\mathrm{Am}(M/F)", "parent_index": 4, "subtype": "inline"}, {"bbox": [223, 493, 232, 500], "content": "F", "parent_index": 5, "subtype": "inline"}, {"bbox": [149, 504, 213, 514], "content": "\\#\\operatorname{Am}(M/F)\\mid", "parent_index": 5, "subtype": "inline"}, {"bbox": [288, 504, 321, 514], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 5, "subtype": "inline"}, {"bbox": [317, 524, 326, 532], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 535, 277, 545], "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "parent_index": 6, "subtype": "inline"}, {"bbox": [348, 535, 452, 545], "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 547, 181, 555], "content": "F", "parent_index": 6, "subtype": "inline"}, {"bbox": [304, 547, 314, 556], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [425, 571, 433, 578], "content": "F", "parent_index": 7, "subtype": "inline"}, {"bbox": [283, 583, 293, 592], "content": "d_{3}", "parent_index": 7, "subtype": "inline"}, {"bbox": [343, 583, 366, 593], "content": "M/F", "parent_index": 7, "subtype": "inline"}, {"bbox": [425, 582, 485, 593], "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "parent_index": 7, "subtype": "inline"}, {"bbox": [150, 598, 156, 603], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [238, 594, 257, 604], "content": "d_{3}y^{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [392, 595, 402, 604], "content": "d_{3}", "parent_index": 7, "subtype": "inline"}, {"bbox": [418, 595, 426, 603], "content": "F", "parent_index": 7, "subtype": "inline"}, {"bbox": [217, 610, 223, 614], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 608, 352, 614], "content": "t\\ =\\ 2", "parent_index": 7, "subtype": "inline"}, {"bbox": [125, 630, 262, 641], "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "parent_index": 7, "subtype": "inline"}, {"bbox": [287, 630, 320, 641], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 7, "subtype": "inline"}, {"bbox": [232, 643, 240, 650], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [303, 643, 333, 652], "content": "\\alpha\\,\\in\\,k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [383, 642, 423, 652], "content": "d_{3}y^{2}\\,<\\,0", "parent_index": 8, "subtype": "inline"}, {"bbox": [439, 654, 462, 665], "content": "M/F", "parent_index": 8, "subtype": "inline"}, {"bbox": [331, 667, 340, 676], "content": "d_{3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [412, 666, 435, 677], "content": "M/F", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 678, 246, 689], "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "parent_index": 8, "subtype": "inline"}, {"bbox": [281, 679, 289, 686], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [477, 679, 485, 686], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [297, 116, 304, 123], "content": "\\mathbb{Z}", "parent_index": 8, "subtype": "inline"}, {"bbox": [358, 115, 410, 126], "content": "(E:H)\\geq4", "parent_index": 8, "subtype": "inline"}, {"bbox": [311, 127, 391, 138], "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "parent_index": 8, "subtype": "inline"}]
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[]
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Next we derive some relations between the class groups of $K_{2}$ and ${\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1.
Proposition 8. Let $L$ and $\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\widetilde{L}$ , respectively, which are not normal o ver $\mathbb{Q}$ .
a) We have 4 $|\mathit{\Omega}_{h}(K_{2})$ if and only if $4\mid h(\widetilde{K}_{2})$ ;
b) $I f\,4\mid h(K_{2})$ , then one of $\mathrm{Cl}_{2}(K_{2})$ or $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\geq4$ .
Proof. Notice that the prime dividing $\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\mathrm{disc}(k_{1})$ .
If we write $K_{2}=k_{2}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{2}$ , then $\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ . In fact, $K_{2}$ and ${\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties
1. $F/k_{2}$ is a quadratic extension unramified outside $\mathfrak{p}$ ;
Therefore it suffices to observe that if $k_{2}(\sqrt{\alpha}\,)$ has these properties, then so does $k_{2}(\sqrt{d_{2}\alpha}\,)$ . But this is elementary.
In particular, the compositum $M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\widetilde{K}_{2}$ and $F=k_{2}(\sqrt{d_{2}}\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\tilde{K}}_{2}$ . Thus $2\parallel h(K_{2})$ implies that $2\parallel h(\widetilde{K}_{2})$ . This proves part a) of the proposition.
Before we go on, we give a Hasse diagram for the fields occurring in this proof:
Now assume that $4\,\mid\,h(K_{2})$ . Since $\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\,=\,\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\,\simeq\,\mathrm{Gal}(N/F)$ : in fact, if $\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\mathrm{Gal}(N/K_{2})$ and $\mathrm{Gal}(N/\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove.
So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ of $F$ dividing $\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since
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<html><body>
<p data-bbox="125 145 486 169">Next we derive some relations between the class groups of $K_{2}$ and ${\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. </p>
<p data-bbox="124 174 486 212">Proposition 8. Let $L$ and $\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\widetilde{L}$ , respectively, which are not normal o ver $\mathbb{Q}$ . </p>
<p data-bbox="133 214 487 253">a) We have 4 $|\mathit{\Omega}_{h}(K_{2})$ if and only if $4\mid h(\widetilde{K}_{2})$ ;
b) $I f\,4\mid h(K_{2})$ , then one of $\mathrm{Cl}_{2}(K_{2})$ or $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\geq4$ . </p>
<p data-bbox="125 259 486 284">Proof. Notice that the prime dividing $\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\mathrm{disc}(k_{1})$ . </p>
<p data-bbox="125 285 486 311">If we write $K_{2}=k_{2}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{2}$ , then $\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ . In fact, $K_{2}$ and ${\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties </p>
<p data-bbox="124 311 363 322">1. $F/k_{2}$ is a quadratic extension unramified outside $\mathfrak{p}$ ; </p>
<p data-bbox="124 334 487 358">Therefore it suffices to observe that if $k_{2}(\sqrt{\alpha}\,)$ has these properties, then so does $k_{2}(\sqrt{d_{2}\alpha}\,)$ . But this is elementary. </p>
<p data-bbox="124 358 487 434">In particular, the compositum $M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\widetilde{K}_{2}$ and $F=k_{2}(\sqrt{d_{2}}\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\tilde{K}}_{2}$ . Thus $2\parallel h(K_{2})$ implies that $2\parallel h(\widetilde{K}_{2})$ . This proves part a) of the proposition. </p>
<p data-bbox="132 434 485 447">Before we go on, we give a Hasse diagram for the fields occurring in this proof: </p>
<div class="image" data-bbox="253 460 376 545"><img data-bbox="253 460 376 545"/></div>
<p data-bbox="124 554 487 663">Now assume that $4\,\mid\,h(K_{2})$ . Since $\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\,=\,\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\,\simeq\,\mathrm{Gal}(N/F)$ : in fact, if $\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\mathrm{Gal}(N/K_{2})$ and $\mathrm{Gal}(N/\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. </p>
<p data-bbox="124 663 486 700">So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ of $F$ dividing $\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since </p>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 9}, {"type": "text", "text": "Next we derive some relations between the class groups of $K_{2}$ and ${\\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. ", "page_idx": 9}, {"type": "text", "text": "Proposition 8. Let $L$ and $\\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\\widetilde{L}$ , respectively, which are not normal o ver $\\mathbb{Q}$ . ", "page_idx": 9}, {"type": "text", "text": "a) We have 4 $|\\mathit{\\Omega}_{h}(K_{2})$ if and only if $4\\mid h(\\widetilde{K}_{2})$ ; \nb) $I f\\,4\\mid h(K_{2})$ , then one of $\\mathrm{Cl}_{2}(K_{2})$ or $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\\geq4$ . ", "page_idx": 9}, {"type": "text", "text": "Proof. Notice that the prime dividing $\\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\\mathrm{disc}(k_{1})$ . ", "page_idx": 9}, {"type": "text", "text": "If we write $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{2}$ , then $\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ . In fact, $K_{2}$ and ${\\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties ", "page_idx": 9}, {"type": "text", "text": "1. $F/k_{2}$ is a quadratic extension unramified outside $\\mathfrak{p}$ ; ", "page_idx": 9}, {"type": "text", "text": "Therefore it suffices to observe that if $k_{2}(\\sqrt{\\alpha}\\,)$ has these properties, then so does $k_{2}(\\sqrt{d_{2}\\alpha}\\,)$ . But this is elementary. ", "page_idx": 9}, {"type": "text", "text": "In particular, the compositum $M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\\widetilde{K}_{2}$ and $F=k_{2}(\\sqrt{d_{2}}\\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\\tilde{K}}_{2}$ . Thus $2\\parallel h(K_{2})$ implies that $2\\parallel h(\\widetilde{K}_{2})$ . This proves part a) of the proposition. ", "page_idx": 9}, {"type": "text", "text": "Before we go on, we give a Hasse diagram for the fields occurring in this proof: ", "page_idx": 9}, {"type": "image", "img_path": "images/d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "Now assume that $4\\,\\mid\\,h(K_{2})$ . Since $\\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\\,=\\,\\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)$ : in fact, if $\\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\\mathrm{Gal}(N/K_{2})$ and $\\mathrm{Gal}(N/\\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. ", "page_idx": 9}, {"type": "text", "text": "So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ of $F$ dividing $\\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since ", "page_idx": 9}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [{"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text"}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text"}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text"}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text"}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "口", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [126, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [], "index": 0.5, "bbox_fs": [125, 114, 486, 138], "lines_deleted": true}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5, "bbox_fs": [125, 146, 487, 170]}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "bbox_fs": [126, 177, 486, 214]}, {"type": "list", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8, "is_list_start_line": true}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9, "is_list_end_line": true}], "index": 8, "bbox_fs": [135, 216, 486, 254]}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5, "bbox_fs": [125, 262, 486, 286]}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [126, 285, 484, 312]}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [126, 311, 363, 325]}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [125, 335, 486, 360]}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5, "bbox_fs": [124, 360, 488, 436]}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [137, 436, 484, 449]}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30, "bbox_fs": [125, 556, 487, 665]}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [125, 665, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 487, 137], "content": "", "index": 0}, {"type": "text", "bbox": [125, 145, 486, 169], "content": "Next we derive some relations between the class groups of and ; these relations will allow us to use each of them as our field in Theorem 1.", "index": 1}, {"type": "text", "bbox": [124, 174, 486, 212], "content": "Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver .", "index": 2}, {"type": "list", "bbox": [133, 214, 487, 253], "content": "", "index": 3}, {"type": "text", "bbox": [125, 259, 486, 284], "content": "Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing .", "index": 4}, {"type": "text", "bbox": [125, 285, 486, 311], "content": "If we write for some , then . In fact, and are the only extensions of with the p roperties", "index": 5}, {"type": "text", "bbox": [124, 311, 363, 322], "content": "1. is a quadratic extension unramified outside ;", "index": 6}, {"type": "text", "bbox": [124, 334, 487, 358], "content": "Therefore it suffices to observe that if has these properties, then so does . But this is elementary.", "index": 7}, {"type": "text", "bbox": [124, 358, 487, 434], "content": "In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition.", "index": 8}, {"type": "text", "bbox": [132, 434, 485, 447], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "index": 9}, {"type": "image", "bbox": [253, 460, 376, 545], "content": "", "index": 10}, {"type": "text", "bbox": [124, 554, 487, 663], "content": "Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove.", "index": 11}, {"type": "text", "bbox": [124, 663, 486, 700], "content": "So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since", "index": 12}]
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[{"bbox": [136, 146, 487, 159], "content": "Next we derive some relations between the class groups of and ; these", "parent_index": 1, "line_index": 0}, {"bbox": [125, 159, 439, 170], "content": "relations will allow us to use each of them as our field in Theorem 1.", "parent_index": 1, "line_index": 1}, {"bbox": [126, 177, 485, 190], "content": "Proposition 8. Let and be the two unramified cyclic quartic extensions of ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 189, 486, 203], "content": "and let and be two qu a dratic extensions of in and , respectively, which", "parent_index": 2, "line_index": 1}, {"bbox": [126, 203, 228, 214], "content": "are not normal o ver .", "parent_index": 2, "line_index": 2}, {"bbox": [136, 216, 340, 229], "content": "a) We have 4 if and only if ;", "parent_index": 3, "line_index": 0}, {"bbox": [135, 230, 486, 243], "content": "b) , then one of or has type , whereas the other", "parent_index": 3, "line_index": 1}, {"bbox": [150, 243, 246, 254], "content": "is cyclic of order .", "parent_index": 3, "line_index": 2}, {"bbox": [126, 262, 486, 274], "content": "Proof. Notice that the prime dividing splits in . Throughout this proof,", "parent_index": 4, "line_index": 0}, {"bbox": [125, 273, 342, 286], "content": "let be one of the primes of dividing .", "parent_index": 4, "line_index": 1}, {"bbox": [136, 285, 484, 299], "content": "If we write for some , then . In fact,", "parent_index": 5, "line_index": 0}, {"bbox": [126, 299, 399, 312], "content": "and are the only extensions of with the p roperties", "parent_index": 5, "line_index": 1}, {"bbox": [126, 311, 363, 325], "content": "1. is a quadratic extension unramified outside ;", "parent_index": 6, "line_index": 0}, {"bbox": [125, 335, 486, 348], "content": "Therefore it suffices to observe that if has these properties, then so does", "parent_index": 7, "line_index": 0}, {"bbox": [126, 347, 277, 360], "content": ". But this is elementary.", "parent_index": 7, "line_index": 1}, {"bbox": [137, 360, 488, 374], "content": "In particular, the compositum is an extension of", "parent_index": 8, "line_index": 0}, {"bbox": [125, 373, 486, 387], "content": "type over with subextensions , and . Clearly is the", "parent_index": 8, "line_index": 1}, {"bbox": [126, 387, 487, 400], "content": "unramified quadratic extension of , so bo t h and are unramified. If", "parent_index": 8, "line_index": 2}, {"bbox": [126, 399, 486, 411], "content": "had 2-class number 2, then would have odd class numb e r, and would also", "parent_index": 8, "line_index": 3}, {"bbox": [124, 411, 486, 425], "content": "be the 2-class field of . Thus implies that . This proves part", "parent_index": 8, "line_index": 4}, {"bbox": [125, 424, 220, 436], "content": "a) of the proposition.", "parent_index": 8, "line_index": 5}, {"bbox": [137, 436, 484, 449], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "parent_index": 9, "line_index": 0}, {"bbox": [137, 556, 487, 569], "content": "Now assume that . Since is cyclic by Lemma 6, there is a", "parent_index": 11, "line_index": 0}, {"bbox": [126, 569, 486, 580], "content": "unique quadratic unramified extension , and the uniqueness implies at once", "parent_index": 11, "line_index": 1}, {"bbox": [125, 579, 487, 593], "content": "that is normal. Hence is a group of order 8 containing a", "parent_index": 11, "line_index": 2}, {"bbox": [126, 592, 486, 604], "content": "subgroup of type : in fact, if were cyclic, then the", "parent_index": 11, "line_index": 3}, {"bbox": [126, 605, 486, 616], "content": "primes ramifying in would also ramify in contradicting the fact that", "parent_index": 11, "line_index": 4}, {"bbox": [126, 616, 485, 628], "content": "is unramified. There are three groups satisfying these conditions: ,", "parent_index": 11, "line_index": 5}, {"bbox": [126, 628, 486, 640], "content": "and . We claim that is non-abelian; once we have proved", "parent_index": 11, "line_index": 6}, {"bbox": [125, 640, 485, 654], "content": "this, it follows that exactly one of the groups and is cyclic,", "parent_index": 11, "line_index": 7}, {"bbox": [125, 653, 383, 665], "content": "and that the other is not, which is what we want to prove.", "parent_index": 11, "line_index": 8}, {"bbox": [137, 665, 487, 678], "content": "So assume that is abelian. Then is ramified at two finite primes and", "parent_index": 12, "line_index": 0}, {"bbox": [126, 677, 485, 690], "content": "of dividing (in ); if and denote the quadratic subextensions of", "parent_index": 12, "line_index": 1}, {"bbox": [125, 689, 486, 702], "content": "different from then and must be ramified at a finite prime (since", "parent_index": 12, "line_index": 2}]
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[{"bbox": [253, 460, 376, 545], "content": "", "parent_index": 10, "subtype": "body"}]
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[265, 413, 307, 424], "content": "2\\parallel h(K_{2})", "parent_index": 8, "subtype": "inline"}, {"bbox": [364, 411, 406, 424], "content": "2\\parallel h(\\widetilde{K}_{2})", "parent_index": 8, "subtype": "inline"}, {"bbox": [220, 558, 263, 568], "content": "4\\,\\mid\\,h(K_{2})", "parent_index": 11, "subtype": "inline"}, {"bbox": [301, 558, 334, 568], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 11, "subtype": "inline"}, {"bbox": [299, 569, 323, 580], "content": "N/M", "parent_index": 11, "subtype": "inline"}, {"bbox": [148, 581, 171, 592], "content": "N/k_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [258, 581, 329, 592], "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "parent_index": 11, "subtype": "inline"}, {"bbox": [206, 593, 288, 604], "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "parent_index": 11, "subtype": "inline"}, {"bbox": [343, 593, 388, 604], "content": "\\operatorname{Gal}(N/F)", "parent_index": 11, "subtype": "inline"}, {"bbox": [217, 605, 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{"bbox": [276, 690, 300, 701], "content": "F_{2}/F", "parent_index": 12, "subtype": "inline"}]
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[]
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$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\mathfrak{q}$ implies ramification at the conjugated ideal ${\mathfrak{q}}^{\prime}$ . Hence both $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified.
We have proved that $\mathrm{Cl_{2}}(K_{2})$ and $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\widetilde{K}}_{2}$ is even, it is sufficient to show that $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\ge2$ and $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$ as claimed.
Now P roposition 5 implies that $\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\geq4$ , and that $\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$ $(2,2)$ . This concludes our proof. 口
Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write
Then $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ).
Proof. Observe that $\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives
$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}
$$
in case A) and
$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}
$$
in case B). Multiplying them together and plugging in the class number formula for $K/\mathbb{Q}$ yields
$$
h_{2}(L)^{2}=\frac{q_{1}\,q_{2}}{8}\frac{h_{2}(K_{1})^{2}\,h_{2}(K_{2})^{2}\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\,h_{2}(k_{2})^{2}}.
$$
Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ as claimed. 口
# 5. Classification
In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.
Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities:
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<html><body>
<p data-bbox="125 111 486 172">$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\mathfrak{q}$ implies ramification at the conjugated ideal ${\mathfrak{q}}^{\prime}$ . Hence both $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. </p>
<p data-bbox="125 173 486 275">We have proved that $\mathrm{Cl_{2}}(K_{2})$ and $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\widetilde{K}}_{2}$ is even, it is sufficient to show that $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\ge2$ and $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$ as claimed. </p>
<p data-bbox="125 276 486 301">Now P roposition 5 implies that $\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\geq4$ , and that $\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$ $(2,2)$ . This concludes our proof. 口 </p>
<p data-bbox="125 310 487 346">Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write </p>
<p data-bbox="126 380 465 394">Then $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). </p>
<p data-bbox="125 401 486 426">Proof. Observe that $\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives </p>
<div class="equation" data-bbox="199 433 412 459">$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}
$$</div>
<p data-bbox="124 463 191 475">in case A) and </p>
<div class="equation" data-bbox="199 484 412 509">$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}
$$</div>
<p data-bbox="125 514 486 538">in case B). Multiplying them together and plugging in the class number formula for $K/\mathbb{Q}$ yields </p>
<div class="equation" data-bbox="219 542 392 568">$$
h_{2}(L)^{2}=\frac{q_{1}\,q_{2}}{8}\frac{h_{2}(K_{1})^{2}\,h_{2}(K_{2})^{2}\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\,h_{2}(k_{2})^{2}}.
$$</div>
<p data-bbox="124 570 487 595">Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ as claimed. 口 </p>
<h1 data-bbox="261 610 349 623">5. Classification </h1>
<p data-bbox="125 629 486 653">In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. </p>
<p data-bbox="124 662 486 700">Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\\mathfrak{q}$ implies ramification at the conjugated ideal ${\\mathfrak{q}}^{\\prime}$ . Hence both $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. ", "page_idx": 10}, {"type": "text", "text": "We have proved that $\\mathrm{Cl_{2}}(K_{2})$ and $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\\widetilde{K}}_{2}$ is even, it is sufficient to show that $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\\ge2$ and $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2$ as claimed. ", "page_idx": 10}, {"type": "text", "text": "Now P roposition 5 implies that $\\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\\geq4$ , and that $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq$ $(2,2)$ . This concludes our proof. 口 ", "page_idx": 10}, {"type": "text", "text": "Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write ", "page_idx": 10}, {"type": "text", "text": "Then $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). ", "page_idx": 10}, {"type": "text", "text": "Proof. Observe that $\\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case A) and ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case B). Multiplying them together and plugging in the class number formula for $K/\\mathbb{Q}$ yields ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ as claimed. 口 ", "page_idx": 10}, {"type": "text", "text": "5. Classification ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. ", "page_idx": 10}, {"type": "text", "text": "Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: ", "page_idx": 10}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "口", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda’s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "口", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 485, 99], "lines": [{"bbox": [476, 93, 486, 101], "spans": [{"bbox": [476, 93, 486, 101], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [125, 114, 486, 174]}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5, "bbox_fs": [125, 174, 487, 277]}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "口", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [126, 277, 487, 302]}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [125, 311, 486, 348]}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [127, 382, 464, 396]}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda’s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 403, 486, 428]}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [124, 465, 191, 477]}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [125, 516, 486, 541]}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "口", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [125, 572, 487, 596]}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [124, 631, 487, 654]}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33, "bbox_fs": [126, 665, 487, 702]}]}
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[{"type": "text", "bbox": [125, 111, 486, 172], "content": "has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified.", "index": 0}, {"type": "text", "bbox": [125, 173, 486, 275], "content": "We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed.", "index": 1}, {"type": "text", "bbox": [125, 276, 486, 301], "content": "Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口", "index": 2}, {"type": "text", "bbox": [125, 310, 487, 346], "content": "Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write", "index": 3}, {"type": "text", "bbox": [126, 380, 465, 394], "content": "Then unless possibly when in case ).", "index": 4}, {"type": "text", "bbox": [125, 401, 486, 426], "content": "Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives", "index": 5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "content": "", "index": 6}, {"type": "text", "bbox": [124, 463, 191, 475], "content": "in case A) and", "index": 7}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "content": "", "index": 8}, {"type": "text", "bbox": [125, 514, 486, 538], "content": "in case B). Multiplying them together and plugging in the class number formula for yields", "index": 9}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "content": "", "index": 10}, {"type": "text", "bbox": [124, 570, 487, 595], "content": "Now , and (by Proposition 6), and taking the square root we find as claimed. 口", "index": 11}, {"type": "title", "bbox": [261, 610, 349, 623], "content": "5. Classification", "index": 12}, {"type": "text", "bbox": [125, 629, 486, 653], "content": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.", "index": 13}, {"type": "text", "bbox": [124, 662, 486, 700], "content": "Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:", "index": 14}]
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[{"bbox": [126, 114, 484, 127], "content": "has odd class number in the strict sense: see Lemma 6); since both and", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 139], "content": "are normal (even abelian) over , ramification at implies ramification at the", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 485, 150], "content": "conjugated ideal . Hence both and ramify in and , and since they", "parent_index": 0, "line_index": 2}, {"bbox": [125, 150, 486, 163], "content": "also ramify in , they must ramify completely in , again contradicting the", "parent_index": 0, "line_index": 3}, {"bbox": [126, 162, 255, 174], "content": "fact that is unramified.", "parent_index": 0, "line_index": 4}, {"bbox": [137, 174, 487, 188], "content": "We have proved that and contain subgroups of type (4) and", "parent_index": 1, "line_index": 0}, {"bbox": [126, 187, 486, 200], "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "parent_index": 1, "line_index": 1}, {"bbox": [126, 199, 486, 212], "content": ". Since the class number of is even, it is sufficient to show that", "parent_index": 1, "line_index": 2}, {"bbox": [126, 212, 487, 227], "content": ". In case A), there is e xactly one ramified prime (it divides", "parent_index": 1, "line_index": 3}, {"bbox": [126, 226, 486, 238], "content": "), hen c e . In case B), there are two ramified", "parent_index": 1, "line_index": 4}, {"bbox": [126, 239, 486, 252], "content": "primes (one is infin i te, the other divides ), hence ;", "parent_index": 1, "line_index": 5}, {"bbox": [125, 251, 487, 264], "content": "but is not a norm residue at the ramified infinite prime, h e nce and", "parent_index": 1, "line_index": 6}, {"bbox": [126, 264, 262, 277], "content": "as claimed.", "parent_index": 1, "line_index": 7}, {"bbox": [137, 277, 487, 290], "content": "Now P roposition 5 implies that is cyclic of order , and that", "parent_index": 2, "line_index": 0}, {"bbox": [126, 290, 487, 302], "content": ". This concludes our proof. 口", "parent_index": 2, "line_index": 1}, {"bbox": [125, 311, 486, 326], "content": "Proposition 9. Assume that is one of the imaginary quadratic fields of type", "parent_index": 3, "line_index": 0}, {"bbox": [126, 324, 486, 337], "content": "or ) as explained in the Introduction. Then there exist two unramified cyclic", "parent_index": 3, "line_index": 1}, {"bbox": [126, 337, 374, 348], "content": "quartic extensions of . Let be one of them, and write", "parent_index": 3, "line_index": 2}, {"bbox": [127, 382, 464, 396], "content": "Then unless possibly when in case ).", "parent_index": 4, "line_index": 0}, {"bbox": [127, 403, 486, 415], "content": "Proof. Observe that in case A) and B); Kuroda’s class number formulas for", "parent_index": 5, "line_index": 0}, {"bbox": [126, 415, 218, 428], "content": "and gives", "parent_index": 5, "line_index": 1}, {"bbox": [124, 465, 191, 477], "content": "in case A) and", "parent_index": 7, "line_index": 0}, {"bbox": [125, 516, 486, 529], "content": "in case B). Multiplying them together and plugging in the class number formula", "parent_index": 9, "line_index": 0}, {"bbox": [126, 528, 192, 541], "content": "for yields", "parent_index": 9, "line_index": 1}, {"bbox": [125, 572, 487, 586], "content": "Now , and (by Proposition 6), and taking the square", "parent_index": 11, "line_index": 0}, {"bbox": [126, 585, 486, 596], "content": "root we find as claimed. 口", "parent_index": 11, "line_index": 1}, {"bbox": [262, 612, 349, 625], "content": "5. Classification", "parent_index": 12, "line_index": 0}, {"bbox": [136, 631, 487, 643], "content": "In this section we apply the results obtained in the last few sections to give a", "parent_index": 13, "line_index": 0}, {"bbox": [124, 644, 218, 654], "content": "proof for Theorem 1.", "parent_index": 13, "line_index": 1}, {"bbox": [126, 665, 487, 677], "content": "Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions", "parent_index": 14, "line_index": 0}, {"bbox": [126, 677, 486, 689], "content": "of , and let be the subgroup of fixing . Then satisfies the", "parent_index": 14, "line_index": 1}, {"bbox": [126, 690, 458, 702], "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "parent_index": 14, "line_index": 2}]
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[]
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[{"bbox": [126, 116, 134, 123], "content": "F", "parent_index": 0, "subtype": "inline"}, {"bbox": [439, 116, 450, 125], "content": "F_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [474, 116, 484, 125], "content": "F_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [269, 128, 279, 137], "content": "k_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [356, 130, 361, 137], "content": "\\mathfrak{q}", "parent_index": 0, "subtype": "inline"}, {"bbox": [200, 139, 208, 149], "content": "{\\mathfrak{q}}^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [267, 142, 272, 149], "content": "\\mathfrak{q}", "parent_index": 0, "subtype": "inline"}, {"bbox": [294, 139, 302, 149], "content": "{\\mathfrak{q}}^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [347, 139, 371, 149], "content": "F_{1}/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 139, 417, 150], "content": "F_{2}/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [189, 151, 212, 162], "content": "M/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [357, 151, 378, 162], "content": "N/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [167, 163, 191, 174], "content": "N/M", "parent_index": 0, "subtype": "inline"}, {"bbox": [235, 176, 270, 187], "content": "\\mathrm{Cl_{2}}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [294, 174, 329, 187], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 188, 148, 199], "content": "(2,2)", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 199, 192, 212], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [314, 199, 327, 211], "content": "{\\widetilde{K}}_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 212, 214, 225], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 228, 136, 237], "content": "d_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [175, 226, 324, 238], "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "parent_index": 1, "subtype": "inline"}, {"bbox": [308, 242, 317, 251], "content": "d_{3}", "parent_index": 1, "subtype": "inline"}, {"bbox": [356, 239, 483, 252], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "parent_index": 1, "subtype": "inline"}, {"bbox": [144, 254, 157, 262], "content": "^{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [414, 253, 466, 263], "content": "(E:H)\\ge2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 264, 210, 277], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "parent_index": 1, "subtype": "inline"}, {"bbox": [272, 280, 308, 290], "content": "\\mathrm{Cl}_{2}(K_{2})", "parent_index": 2, "subtype": "inline"}, {"bbox": [380, 281, 396, 289], "content": "\\geq4", "parent_index": 2, "subtype": "inline"}, {"bbox": [439, 277, 487, 290], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 291, 148, 302], "content": "(2,2)", "parent_index": 2, "subtype": "inline"}, {"bbox": [261, 314, 268, 322], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 325, 138, 336], "content": "A)", "parent_index": 3, "subtype": "inline"}, {"bbox": [154, 326, 162, 334], "content": "B", "parent_index": 3, "subtype": "inline"}, {"bbox": [219, 338, 225, 345], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [250, 338, 257, 345], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [152, 383, 282, 396], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "parent_index": 4, "subtype": "inline"}, {"bbox": [377, 385, 413, 394], "content": "d_{3}=-4", "parent_index": 4, "subtype": "inline"}, {"bbox": [450, 385, 457, 392], "content": "B", "parent_index": 4, "subtype": "inline"}, {"bbox": [217, 405, 242, 412], "content": "\\upsilon=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 416, 147, 426], "content": "L/k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 416, 192, 426], "content": "L/k_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [199, 433, 412, 459], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "parent_index": 6, "subtype": "interline"}, {"bbox": [199, 484, 412, 509], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "parent_index": 8, "subtype": "interline"}, {"bbox": [141, 529, 163, 540], "content": "K/\\mathbb{Q}", "parent_index": 9, "subtype": "inline"}, {"bbox": [219, 542, 392, 568], "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "parent_index": 10, "subtype": "interline"}, {"bbox": [148, 573, 194, 584], "content": "h_{2}(k_{1})=1", "parent_index": 11, "subtype": "inline"}, {"bbox": [199, 573, 246, 584], "content": "h_{2}(k_{2})=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [267, 574, 303, 583], "content": "q_{1}q_{2}=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [182, 585, 312, 596], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "parent_index": 11, "subtype": "inline"}, {"bbox": [237, 667, 244, 674], "content": "L", "parent_index": 14, "subtype": "inline"}, {"bbox": [139, 679, 145, 686], "content": "k", "parent_index": 14, "subtype": "inline"}, {"bbox": [189, 679, 199, 686], "content": "N", "parent_index": 14, "subtype": "inline"}, {"bbox": [295, 677, 339, 689], "content": "\\operatorname{Gal}(k^{2}/k)", "parent_index": 14, "subtype": "inline"}, {"bbox": [372, 679, 379, 686], "content": "L", "parent_index": 14, "subtype": "inline"}, {"bbox": [419, 679, 428, 686], "content": "N", "parent_index": 14, "subtype": "inline"}]
|
[]
|
|
<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>
Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\prime})\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even).
We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.
a) If $h_{2}(K_{2})\,=\,2$ , then $\#\kappa_{2}\,=\,2$ by Proposition 5, hence $q_{2}\,=\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ .
It can be shown using the ambiguous class number formula that $\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1].
b) If $h_{2}(K_{2})=4$ , we may assume that $\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\#\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ .
Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\neq4$ , and by the table at the beginning of this proof this implies that rank $\mathrm{Cl}_{2}(k^{1})\neq2$ in case A).
Next we consider case B); here we have to distinguish between $d_{3}\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ).
Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\#\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\#\kappa_{2}\,=\,2$ , $q_{2}\,=\,2$ and $q_{1}\,=\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ .
Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$ , we get $\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$ . From $K_{2}=k_{2}(\sqrt{\varepsilon}\,)$ (for a suitable choice of $L$ ; the other possibility is ${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$ , where $\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\geq2$ . Moreover, we have $K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$ , where $\pi,\lambda\,\equiv\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\,=\,\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\sqrt{\pi}\,)/K_{1}$ is easily seen to be unramified.
Thus $2\mid q_{1},\,2\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\cdot2\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\prime})\geq3$ in this case.
This concludes the proof.
|
<html><body>
<div class="table" data-bbox="228 110 383 167"><table data-bbox="228 110 383 167"><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></div>
<p data-bbox="124 187 486 248">Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\prime})\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). </p>
<p data-bbox="124 263 485 298">We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. </p>
<p data-bbox="124 299 486 334">a) If $h_{2}(K_{2})\,=\,2$ , then $\#\kappa_{2}\,=\,2$ by Proposition 5, hence $q_{2}\,=\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . </p>
<p data-bbox="125 335 486 382">It can be shown using the ambiguous class number formula that $\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. </p>
<p data-bbox="125 382 485 418">b) If $h_{2}(K_{2})=4$ , we may assume that $\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\#\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . </p>
<p data-bbox="124 418 486 442">Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\neq4$ , and by the table at the beginning of this proof this implies that rank $\mathrm{Cl}_{2}(k^{1})\neq2$ in case A). </p>
<p data-bbox="127 457 486 481">Next we consider case B); here we have to distinguish between $d_{3}\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). </p>
<p data-bbox="124 484 487 553">Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\#\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\#\kappa_{2}\,=\,2$ , $q_{2}\,=\,2$ and $q_{1}\,=\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . </p>
<p data-bbox="125 564 486 663">Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$ , we get $\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$ . From $K_{2}=k_{2}(\sqrt{\varepsilon}\,)$ (for a suitable choice of $L$ ; the other possibility is ${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$ , where $\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\geq2$ . Moreover, we have $K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$ , where $\pi,\lambda\,\equiv\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\,=\,\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\sqrt{\pi}\,)/K_{1}$ is easily seen to be unramified. </p>
<p data-bbox="125 663 485 687">Thus $2\mid q_{1},\,2\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\cdot2\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\prime})\geq3$ in this case. </p>
<p data-bbox="137 688 249 699">This concludes the proof. </p>
</body></html>
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0003244v1
| 11 | 612 | 792 | 1,275 | 1,650 |
[{"type": "table", "img_path": "images/295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>\n\n", "page_idx": 11}, {"type": "text", "text": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\\prime})\\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). ", "page_idx": 11}, {"type": "text", "text": "We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. ", "page_idx": 11}, {"type": "text", "text": "a) If $h_{2}(K_{2})\\,=\\,2$ , then $\\#\\kappa_{2}\\,=\\,2$ by Proposition 5, hence $q_{2}\\,=\\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . ", "page_idx": 11}, {"type": "text", "text": "It can be shown using the ambiguous class number formula that $\\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. ", "page_idx": 11}, {"type": "text", "text": "b) If $h_{2}(K_{2})=4$ , we may assume that $\\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\\#\\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . ", "page_idx": 11}, {"type": "text", "text": "Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ , and by the table at the beginning of this proof this implies that rank $\\mathrm{Cl}_{2}(k^{1})\\neq2$ in case A). ", "page_idx": 11}, {"type": "text", "text": "Next we consider case B); here we have to distinguish between $d_{3}\\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). ", "page_idx": 11}, {"type": "text", "text": "Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\\#\\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\\#\\kappa_{2}\\,=\\,2$ , $q_{2}\\,=\\,2$ and $q_{1}\\,=\\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . ", "page_idx": 11}, {"type": "text", "text": "Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})$ , we get $\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}$ . From $K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)$ (for a suitable choice of $L$ ; the other possibility is ${\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))$ , where $\\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\\geq2$ . Moreover, we have $K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})$ , where $\\pi,\\lambda\\,\\equiv\\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\\,=\\,\\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\\sqrt{\\pi}\\,)/K_{1}$ is easily seen to be unramified. ", "page_idx": 11}, {"type": "text", "text": "Thus $2\\mid q_{1},\\,2\\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\\cdot2\\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\\prime})\\geq3$ in this case. ", "page_idx": 11}, {"type": "text", "text": "This concludes the proof. ", "page_idx": 11}]
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{"preproc_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz’s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda’s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [476, 690, 486, 700], "spans": [{"bbox": [476, 690, 486, 700], "score": 0.9837851524353027, "content": "口", "type": "text"}]}]}, {"type": "discarded", "bbox": [125, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7, "bbox_fs": [125, 189, 487, 250]}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [125, 263, 486, 299]}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [124, 300, 487, 336]}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz’s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5, "bbox_fs": [125, 336, 486, 384]}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [124, 383, 486, 421]}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [126, 419, 486, 444]}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [126, 459, 487, 484]}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5, "bbox_fs": [123, 483, 486, 556]}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda’s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5, "bbox_fs": [124, 566, 487, 665]}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5, "bbox_fs": [126, 662, 486, 689]}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43, "bbox_fs": [137, 689, 249, 700]}]}
|
[{"type": "table", "bbox": [228, 110, 383, 167], "content": "", "index": 0}, {"type": "text", "bbox": [124, 187, 486, 248], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even).", "index": 1}, {"type": "text", "bbox": [124, 263, 485, 298], "content": "We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.", "index": 2}, {"type": "text", "bbox": [124, 299, 486, 334], "content": "a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and .", "index": 3}, {"type": "text", "bbox": [125, 335, 486, 382], "content": "It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1].", "index": 4}, {"type": "text", "bbox": [125, 382, 485, 418], "content": "b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and .", "index": 5}, {"type": "text", "bbox": [124, 418, 486, 442], "content": "Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A).", "index": 6}, {"type": "text", "bbox": [127, 457, 486, 481], "content": "Next we consider case B); here we have to distinguish between (case ) and (case ).", "index": 7}, {"type": "text", "bbox": [124, 484, 487, 553], "content": "Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and .", "index": 8}, {"type": "text", "bbox": [125, 564, 486, 663], "content": "Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified.", "index": 9}, {"type": "text", "bbox": [125, 663, 485, 687], "content": "Thus , and so we find that is divisible by . In particular, we always have in this case.", "index": 10}, {"type": "text", "bbox": [137, 688, 249, 699], "content": "This concludes the proof.", "index": 11}]
|
[{"bbox": [125, 189, 486, 201], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "parent_index": 1, "line_index": 0}, {"bbox": [125, 201, 487, 213], "content": "claim to hold if in case B)) is a consequence of the class number formula of", "parent_index": 1, "line_index": 1}, {"bbox": [126, 213, 485, 225], "content": "Proposition 9. In particular, we have if one of the class numbers", "parent_index": 1, "line_index": 2}, {"bbox": [126, 225, 486, 237], "content": "or is at least 8. Therefore it suffices to examine the cases and", "parent_index": 1, "line_index": 3}, {"bbox": [126, 237, 385, 250], "content": "(recall from above that is always even).", "parent_index": 1, "line_index": 4}, {"bbox": [137, 263, 486, 278], "content": "We start by considering case A); it is sufficient to show that .", "parent_index": 2, "line_index": 0}, {"bbox": [125, 275, 486, 290], "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "parent_index": 2, "line_index": 1}, {"bbox": [126, 290, 135, 299], "content": "8.", "parent_index": 2, "line_index": 2}, {"bbox": [125, 300, 487, 313], "content": "a) If , then by Proposition 5, hence by Proposition", "parent_index": 3, "line_index": 0}, {"bbox": [124, 311, 487, 326], "content": "7 and then by Proposition 6. The class number formulas in the proof of", "parent_index": 3, "line_index": 1}, {"bbox": [125, 324, 356, 336], "content": "Proposition 9 now give and .", "parent_index": 3, "line_index": 2}, {"bbox": [137, 336, 486, 348], "content": "It can be shown using the ambiguous class number formula that is trivial", "parent_index": 4, "line_index": 0}, {"bbox": [125, 348, 485, 361], "content": "if and only if is a quadratic nonresidue modulo the prime ideal over in ; by", "parent_index": 4, "line_index": 1}, {"bbox": [126, 361, 486, 373], "content": "Scholz’s reciprocity law, this is equivalent to , and this agrees", "parent_index": 4, "line_index": 2}, {"bbox": [126, 372, 258, 384], "content": "with the criterion given in [1].", "parent_index": 4, "line_index": 3}, {"bbox": [125, 383, 486, 397], "content": "b) If , we may assume that from Proposition 8.b). Then", "parent_index": 5, "line_index": 0}, {"bbox": [126, 397, 486, 409], "content": "by Proposition 5, by Proposition 7 and by Proposition 6.", "parent_index": 5, "line_index": 1}, {"bbox": [124, 407, 435, 421], "content": "Using the class number formula we get and .", "parent_index": 5, "line_index": 2}, {"bbox": [137, 419, 486, 433], "content": "Thus in both cases we have , and by the table at the beginning", "parent_index": 6, "line_index": 0}, {"bbox": [126, 432, 380, 444], "content": "of this proof this implies that rank in case A).", "parent_index": 6, "line_index": 1}, {"bbox": [136, 459, 487, 472], "content": "Next we consider case B); here we have to distinguish between (case", "parent_index": 7, "line_index": 0}, {"bbox": [126, 471, 248, 484], "content": ") and (case ).", "parent_index": 7, "line_index": 1}, {"bbox": [137, 483, 256, 496], "content": "Let us start with case ).", "parent_index": 8, "line_index": 0}, {"bbox": [126, 495, 486, 509], "content": "a) If , then , and as above. The class number", "parent_index": 8, "line_index": 1}, {"bbox": [123, 507, 324, 520], "content": "formula gives and .", "parent_index": 8, "line_index": 2}, {"bbox": [125, 518, 486, 533], "content": "b) If (which we may assume without loss of generality by Proposition", "parent_index": 8, "line_index": 3}, {"bbox": [125, 531, 485, 544], "content": "8.b)) then , and , again exactly as above. This implies", "parent_index": 8, "line_index": 4}, {"bbox": [126, 542, 263, 556], "content": "and .", "parent_index": 8, "line_index": 5}, {"bbox": [125, 566, 487, 580], "content": "Here we apply Kuroda’s class number formula (see [10]) to , and since", "parent_index": 9, "line_index": 0}, {"bbox": [126, 579, 486, 592], "content": "and , we get . From", "parent_index": 9, "line_index": 1}, {"bbox": [126, 592, 485, 605], "content": "(for a suitable choice of ; the other possibility is ,", "parent_index": 9, "line_index": 2}, {"bbox": [126, 604, 487, 617], "content": "where is the fundamental unit of , we deduce that the uni t , which still is", "parent_index": 9, "line_index": 3}, {"bbox": [124, 617, 487, 630], "content": "fundamental in , becomes a square in , and this implies that . Moreover,", "parent_index": 9, "line_index": 4}, {"bbox": [125, 628, 484, 642], "content": "we have , where mod 4 are prime factors of and", "parent_index": 9, "line_index": 5}, {"bbox": [126, 642, 486, 653], "content": "in , respectively. This shows that has even class number, because", "parent_index": 9, "line_index": 6}, {"bbox": [126, 653, 319, 665], "content": "is easily seen to be unramified.", "parent_index": 9, "line_index": 7}, {"bbox": [136, 662, 486, 679], "content": "Thus , and so we find that is divisible by .", "parent_index": 10, "line_index": 0}, {"bbox": [126, 678, 354, 689], "content": "In particular, we always have in this case.", "parent_index": 10, "line_index": 1}, {"bbox": [137, 689, 249, 700], "content": "This concludes the proof.", "parent_index": 11, "line_index": 0}]
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[]
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[{"bbox": [194, 203, 230, 212], "content": "d_{3}=-4", "parent_index": 1, "subtype": "inline"}, {"bbox": [290, 214, 332, 225], "content": "d(G^{\\prime})\\geq3", "parent_index": 1, "subtype": "inline"}, {"bbox": [454, 214, 485, 225], "content": "h_{2}(K_{1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 226, 169, 237], "content": "h_{2}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [415, 226, 465, 237], "content": "h_{2}(K_{2})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 238, 175, 249], "content": "h_{2}(K_{2})=4", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 238, 313, 249], "content": "h_{2}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [402, 266, 482, 276], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "parent_index": 2, "subtype": "inline"}, {"bbox": [150, 302, 202, 312], "content": "h_{2}(K_{2})\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [232, 302, 272, 311], "content": "\\#\\kappa_{2}\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [385, 303, 416, 312], "content": "q_{2}\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [178, 315, 208, 323], "content": "q_{1}=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [229, 325, 279, 336], "content": "h_{2}(K_{1})=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [301, 325, 352, 336], "content": "h_{2}(L)=2^{m}", "parent_index": 3, "subtype": "inline"}, {"bbox": [412, 338, 447, 348], "content": "\\mathrm{Cl}_{2}(K_{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 353, 194, 359], "content": "\\varepsilon_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [434, 350, 444, 359], "content": "d_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [459, 350, 468, 359], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [322, 361, 414, 372], "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [149, 385, 198, 396], "content": "h_{2}(K_{2})=4", "parent_index": 5, "subtype": "inline"}, {"bbox": [296, 385, 358, 396], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 398, 164, 407], "content": "\\#\\kappa_{2}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [247, 398, 276, 407], "content": "q_{2}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [376, 398, 405, 407], "content": "q_{1}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [299, 409, 348, 420], "content": "h_{2}(K_{1})=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [370, 408, 431, 420], "content": "h_{2}(L)=2^{m+2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [257, 421, 337, 432], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "parent_index": 6, "subtype": "inline"}, {"bbox": [279, 433, 329, 443], "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "parent_index": 6, "subtype": "inline"}, {"bbox": [421, 461, 460, 471], "content": "d_{3}\\neq-4", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 473, 138, 482], "content": "B_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [164, 473, 201, 482], "content": "d_{3}=-4", "parent_index": 7, "subtype": "inline"}, {"bbox": [228, 473, 241, 482], "content": "B_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [236, 485, 248, 494], "content": "B_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [149, 496, 200, 507], "content": "h_{2}(K_{2})=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [230, 497, 268, 506], "content": "\\#\\kappa_{2}=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [275, 497, 304, 506], "content": "q_{2}=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [328, 497, 357, 506], "content": "q_{1}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [187, 509, 236, 519], "content": "h_{2}(K_{1})=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [259, 508, 320, 519], "content": "h_{2}(L)=2^{m+1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [147, 520, 208, 531], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "parent_index": 8, "subtype": "inline"}, {"bbox": [175, 533, 216, 542], "content": "\\#\\kappa_{2}\\,=\\,2", "parent_index": 8, "subtype": "inline"}, {"bbox": [223, 533, 254, 542], "content": "q_{2}\\,=\\,2", "parent_index": 8, "subtype": "inline"}, {"bbox": [280, 533, 311, 542], "content": "q_{1}\\,=\\,1", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 544, 175, 555], "content": "h_{2}(K_{1})=4", "parent_index": 8, "subtype": "inline"}, {"bbox": [198, 544, 259, 555], "content": "h_{2}(L)=2^{m+3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [378, 568, 399, 579], "content": "L/k_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [447, 568, 487, 579], "content": "h_{2}(k_{1})=", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 581, 131, 588], "content": "^{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [154, 580, 231, 591], "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "parent_index": 9, "subtype": "inline"}, {"bbox": [269, 579, 453, 591], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 594, 185, 605], "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "parent_index": 9, "subtype": "inline"}, {"bbox": [295, 595, 302, 602], "content": "L", "parent_index": 9, "subtype": "inline"}, {"bbox": [409, 592, 482, 605], "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "parent_index": 9, "subtype": "inline"}, {"bbox": [155, 609, 160, 614], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [286, 607, 297, 615], "content": "k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [416, 609, 421, 614], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [195, 619, 201, 627], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [299, 619, 306, 626], "content": "L", "parent_index": 9, "subtype": "inline"}, {"bbox": [406, 619, 434, 628], "content": "q_{1}\\geq2", "parent_index": 9, "subtype": "inline"}, {"bbox": [166, 629, 236, 641], "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "parent_index": 9, "subtype": "inline"}, {"bbox": [273, 631, 312, 640], "content": "\\pi,\\lambda\\,\\equiv\\,1", "parent_index": 9, "subtype": "inline"}, {"bbox": [440, 631, 450, 640], "content": "d_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [475, 631, 484, 640], "content": "d_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [138, 642, 184, 653], "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "parent_index": 9, "subtype": "inline"}, {"bbox": [327, 643, 340, 652], "content": "K_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 654, 181, 665], "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [163, 666, 235, 677], "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [325, 666, 350, 677], "content": "h_{2}(L)", "parent_index": 10, "subtype": "inline"}, {"bbox": [414, 666, 482, 674], "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 678, 299, 689], "content": "d(G^{\\prime})\\geq3", "parent_index": 10, "subtype": "inline"}]
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The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density.
Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\,=\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$ and $r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\mathfrak{p}}^{h}\,=\,(\pi)$ and $\mathfrak{r}^{h}\,=\,(\rho)$ for primary elements $\pi$ and $\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$ for an ideal $\mathfrak{p}$ with odd norm, then $\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\cal O}_{K}$ ) if and only if $\mathfrak{a}$ is primary (i.e. $[\varepsilon/{\mathfrak a}]=+1$ for all units $\varepsilon\in{\mathcal{O}}_{K}^{\times}$ , where $[\,\cdot\,/\,\cdot\,]$ denotes the quadratic residue symbol in $K$ )). Let $\big[\cdot\big/\cdot\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\pi/\rho]=+1$ and $[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$ .
Put $K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$ and $\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$ ; we claim that $h_{2}(\tilde{K}_{1})\;=\;2$ . This is equivalent to $h_{2}(\widetilde{L}_{1})\,=\,1$ , where $\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$ is a quad r atic unramified extension of $\widetilde{K}_{1}$ . Put $\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$ a n d apply the ambiguous class number formula to $\widetilde{F}_{1}/k_{1}$ an d $\widetilde{L}_{1}/\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$ ; note that we have used the assumption that $[\pi/\rho^{\prime}]=-1$ in deducing th a t ${\mathfrak{r}}^{\prime}$ is inert in $\widetilde{F}_{1}/k_{1}$ .
In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4:
<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>
In order to decide whether $\widetilde{q}_{2}=1$ or $\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\widetilde{L})\,=\,2^{m+2}$ and $\widetilde{q}_{2}=1$ . In particular we see that $4\mid h_{2}(K_{2})$ if and only if $4\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\sqrt{\pi\rho}\,)$ with $[\pi/\rho]=+1$ .
The ambiguous class number formula shows that $\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\sqrt{\pi}\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$ , we see that $2\:\:|\:\:h_{2}(L_{1})$ if and only if $(E\,:\,H)\;=\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\,\equiv\,1\;\mathrm{mod}\;4$ ) and a fundamental unit $\varepsilon$ . Therefore $(E\,:\,H)\;=\;1$ if and only if $\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$ , where $\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$ and where $\{\,\cdot\,/\,\cdot\,\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$ , we have proved that $4\mid h_{2}(K_{1})$ if and only if the prime ideal $\Re_{1}$ above $\mathfrak{r}$ splits in the quadratic extension $F_{1}(\sqrt{\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ .
If $d_{2}=8$ and $p=2$ , then $2\mathcal{O}_{k_{1}}=22^{\prime}$ , and we have to choose $2^{h}=(\pi)$ in such a way that $k_{1}(\sqrt{\pi}\,)/k_{1}$ is unramified outside $\mathfrak{p}$ . The residue symbols $\left[\alpha/2\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\sqrt{\alpha}\,)/k_{1}$ . With these modifactions, the above arguments remain valid.
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<html><body>
<p data-bbox="125 112 486 148">The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. </p>
<p data-bbox="124 149 486 304">Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\,=\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$ and $r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\mathfrak{p}}^{h}\,=\,(\pi)$ and $\mathfrak{r}^{h}\,=\,(\rho)$ for primary elements $\pi$ and $\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$ for an ideal $\mathfrak{p}$ with odd norm, then $\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\cal O}_{K}$ ) if and only if $\mathfrak{a}$ is primary (i.e. $[\varepsilon/{\mathfrak a}]=+1$ for all units $\varepsilon\in{\mathcal{O}}_{K}^{\times}$ , where $[\,\cdot\,/\,\cdot\,]$ denotes the quadratic residue symbol in $K$ )). Let $\big[\cdot\big/\cdot\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\pi/\rho]=+1$ and $[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$ . </p>
<p data-bbox="124 306 487 384">Put $K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$ and $\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$ ; we claim that $h_{2}(\tilde{K}_{1})\;=\;2$ . This is equivalent to $h_{2}(\widetilde{L}_{1})\,=\,1$ , where $\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$ is a quad r atic unramified extension of $\widetilde{K}_{1}$ . Put $\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$ a n d apply the ambiguous class number formula to $\widetilde{F}_{1}/k_{1}$ an d $\widetilde{L}_{1}/\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$ ; note that we have used the assumption that $[\pi/\rho^{\prime}]=-1$ in deducing th a t ${\mathfrak{r}}^{\prime}$ is inert in $\widetilde{F}_{1}/k_{1}$ . </p>
<div class="table" data-bbox="189 412 422 468"><p class="caption" data-bbox="125 385 486 408">In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4: </p><table data-bbox="189 412 422 468"><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></div>
<p data-bbox="125 470 486 520">In order to decide whether $\widetilde{q}_{2}=1$ or $\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\widetilde{L})\,=\,2^{m+2}$ and $\widetilde{q}_{2}=1$ . In particular we see that $4\mid h_{2}(K_{2})$ if and only if $4\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\sqrt{\pi\rho}\,)$ with $[\pi/\rho]=+1$ . </p>
<p data-bbox="124 520 486 651">The ambiguous class number formula shows that $\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\sqrt{\pi}\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$ , we see that $2\:\:|\:\:h_{2}(L_{1})$ if and only if $(E\,:\,H)\;=\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\,\equiv\,1\;\mathrm{mod}\;4$ ) and a fundamental unit $\varepsilon$ . Therefore $(E\,:\,H)\;=\;1$ if and only if $\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$ , where $\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$ and where $\{\,\cdot\,/\,\cdot\,\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$ , we have proved that $4\mid h_{2}(K_{1})$ if and only if the prime ideal $\Re_{1}$ above $\mathfrak{r}$ splits in the quadratic extension $F_{1}(\sqrt{\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . </p>
<p data-bbox="125 651 486 699">If $d_{2}=8$ and $p=2$ , then $2\mathcal{O}_{k_{1}}=22^{\prime}$ , and we have to choose $2^{h}=(\pi)$ in such a way that $k_{1}(\sqrt{\pi}\,)/k_{1}$ is unramified outside $\mathfrak{p}$ . The residue symbols $\left[\alpha/2\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\sqrt{\alpha}\,)/k_{1}$ . With these modifactions, the above arguments remain valid. </p>
</body></html>
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0003244v1
| 12 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. ", "page_idx": 12}, {"type": "text", "text": "Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\\,=\\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}$ and $r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\\mathfrak{p}}^{h}\\,=\\,(\\pi)$ and $\\mathfrak{r}^{h}\\,=\\,(\\rho)$ for primary elements $\\pi$ and $\\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}$ for an ideal $\\mathfrak{p}$ with odd norm, then $\\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\\cal O}_{K}$ ) if and only if $\\mathfrak{a}$ is primary (i.e. $[\\varepsilon/{\\mathfrak a}]=+1$ for all units $\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}$ , where $[\\,\\cdot\\,/\\,\\cdot\\,]$ denotes the quadratic residue symbol in $K$ )). Let $\\big[\\cdot\\big/\\cdot\\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\\pi/\\rho]=+1$ and $[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1$ . ", "page_idx": 12}, {"type": "text", "text": "Put $K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)$ and $\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})$ ; we claim that $h_{2}(\\tilde{K}_{1})\\;=\\;2$ . This is equivalent to $h_{2}(\\widetilde{L}_{1})\\,=\\,1$ , where $\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})$ is a quad r atic unramified extension of $\\widetilde{K}_{1}$ . Put $\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)$ a n d apply the ambiguous class number formula to $\\widetilde{F}_{1}/k_{1}$ an d $\\widetilde{L}_{1}/\\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1$ ; note that we have used the assumption that $[\\pi/\\rho^{\\prime}]=-1$ in deducing th a t ${\\mathfrak{r}}^{\\prime}$ is inert in $\\widetilde{F}_{1}/k_{1}$ . ", "page_idx": 12}, {"type": "table", "img_path": "images/dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg", "table_caption": ["In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4: "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>\n\n", "page_idx": 12}, {"type": "text", "text": "In order to decide whether $\\widetilde{q}_{2}=1$ or $\\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\\widetilde{L})\\,=\\,2^{m+2}$ and $\\widetilde{q}_{2}=1$ . In particular we see that $4\\mid h_{2}(K_{2})$ if and only if $4\\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)$ with $[\\pi/\\rho]=+1$ . ", "page_idx": 12}, {"type": "text", "text": "The ambiguous class number formula shows that $\\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\\sqrt{\\pi}\\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)$ , we see that $2\\:\\:|\\:\\:h_{2}(L_{1})$ if and only if $(E\\,:\\,H)\\;=\\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4$ ) and a fundamental unit $\\varepsilon$ . Therefore $(E\\,:\\,H)\\;=\\;1$ if and only if $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1$ , where $\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}$ and where $\\{\\,\\cdot\\,/\\,\\cdot\\,\\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1$ , we have proved that $4\\mid h_{2}(K_{1})$ if and only if the prime ideal $\\Re_{1}$ above $\\mathfrak{r}$ splits in the quadratic extension $F_{1}(\\sqrt{\\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . ", "page_idx": 12}, {"type": "text", "text": "If $d_{2}=8$ and $p=2$ , then $2\\mathcal{O}_{k_{1}}=22^{\\prime}$ , and we have to choose $2^{h}=(\\pi)$ in such a way that $k_{1}(\\sqrt{\\pi}\\,)/k_{1}$ is unramified outside $\\mathfrak{p}$ . The residue symbols $\\left[\\alpha/2\\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\\sqrt{\\alpha}\\,)/k_{1}$ . With these modifactions, the above arguments remain valid. ", "page_idx": 12}]
|
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{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 |", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 486, 99], "lines": [{"bbox": [476, 92, 487, 101], "spans": [{"bbox": [476, 92, 487, 101], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [126, 114, 486, 150]}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9, "bbox_fs": [125, 150, 487, 307]}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5, "bbox_fs": [125, 307, 486, 386]}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [124, 471, 487, 522]}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 |", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38, "bbox_fs": [125, 521, 487, 654]}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5, "bbox_fs": [125, 652, 487, 702]}]}
|
[{"type": "text", "bbox": [125, 112, 486, 148], "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density.", "index": 0}, {"type": "text", "bbox": [124, 149, 486, 304], "content": "Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and .", "index": 1}, {"type": "text", "bbox": [124, 306, 487, 384], "content": "Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in .", "index": 2}, {"type": "table", "bbox": [189, 412, 422, 468], "content": "", "index": 3}, {"type": "text", "bbox": [125, 470, 486, 520], "content": "In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with .", "index": 4}, {"type": "text", "bbox": [124, 520, 486, 651], "content": "The ambiguous class number formula shows that is cyclic, thus 4 | if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , .", "index": 5}, {"type": "text", "bbox": [125, 651, 486, 699], "content": "If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.", "index": 6}]
|
[{"bbox": [137, 114, 486, 127], "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 485, 138], "content": "whether and infinitely often. Let us show how to prove that", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 313, 150], "content": "both possibilities occur with equal density.", "parent_index": 0, "line_index": 2}, {"bbox": [137, 150, 484, 163], "content": "Before we can do this, we have to study the quadratic extensions and", "parent_index": 1, "line_index": 0}, {"bbox": [125, 161, 487, 176], "content": "of more closely. We assume that and are odd primes in t he", "parent_index": 1, "line_index": 1}, {"bbox": [125, 174, 487, 187], "content": "following, and then say how to modify the arguments in the case or .", "parent_index": 1, "line_index": 2}, {"bbox": [126, 187, 487, 199], "content": "The primes and split in as and . Let denote the", "parent_index": 1, "line_index": 3}, {"bbox": [125, 198, 487, 211], "content": "odd class number of and write and for primary elements", "parent_index": 1, "line_index": 4}, {"bbox": [126, 210, 487, 223], "content": "and (this is can easily be proved directly, but it is also a very special case", "parent_index": 1, "line_index": 5}, {"bbox": [125, 223, 486, 235], "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd", "parent_index": 1, "line_index": 6}, {"bbox": [125, 234, 485, 247], "content": "class number (see [7]): if for an ideal with odd norm, then", "parent_index": 1, "line_index": 7}, {"bbox": [126, 247, 487, 259], "content": "can be chosen primary (i.e. congruent to a square mod ) if and only if is", "parent_index": 1, "line_index": 8}, {"bbox": [125, 258, 487, 272], "content": "primary (i.e. for all units , where denotes the quadratic", "parent_index": 1, "line_index": 9}, {"bbox": [126, 271, 486, 283], "content": "residue symbol in )). Let denote the quadratic residue symbol in . Then", "parent_index": 1, "line_index": 10}, {"bbox": [126, 282, 485, 295], "content": ", so we may choose the conjugates in such a way", "parent_index": 1, "line_index": 11}, {"bbox": [125, 294, 310, 307], "content": "that and .", "parent_index": 1, "line_index": 12}, {"bbox": [136, 307, 486, 320], "content": "Put and ; we claim that . This", "parent_index": 2, "line_index": 0}, {"bbox": [125, 321, 486, 334], "content": "is equivalent to , where is a quad r atic unramified", "parent_index": 2, "line_index": 1}, {"bbox": [126, 333, 486, 349], "content": "extension of . Put a n d apply the ambiguous class number formula", "parent_index": 2, "line_index": 2}, {"bbox": [125, 347, 486, 361], "content": "to an d : since there is only one ramified prime in each of these two", "parent_index": 2, "line_index": 3}, {"bbox": [125, 360, 486, 373], "content": "ext e nsions, w e fin d ; note that we have used the", "parent_index": 2, "line_index": 4}, {"bbox": [126, 374, 415, 386], "content": "assumption that in deducing th a t is inert in .", "parent_index": 2, "line_index": 5}, {"bbox": [124, 471, 486, 486], "content": "In order to decide whether or , recall that we have ; thus", "parent_index": 4, "line_index": 0}, {"bbox": [126, 483, 487, 498], "content": "must be the field with 2 -class nu mber 2, and this implies and", "parent_index": 4, "line_index": 1}, {"bbox": [126, 497, 487, 511], "content": ". In particular we see that if and only if as long as", "parent_index": 4, "line_index": 2}, {"bbox": [126, 510, 268, 522], "content": "with .", "parent_index": 4, "line_index": 3}, {"bbox": [137, 521, 487, 534], "content": "The ambiguous class number formula shows that is cyclic, thus 4 |", "parent_index": 5, "line_index": 0}, {"bbox": [126, 534, 486, 545], "content": "if and only if , where is the quadratic unramified", "parent_index": 5, "line_index": 1}, {"bbox": [125, 545, 487, 558], "content": "extension of . Applying the ambiguous class number formula to , where", "parent_index": 5, "line_index": 2}, {"bbox": [126, 558, 487, 570], "content": ", we see that if and only if . Now is", "parent_index": 5, "line_index": 3}, {"bbox": [125, 569, 487, 582], "content": "generated by a root of unity (which always is a norm residue at primes dividing", "parent_index": 5, "line_index": 4}, {"bbox": [126, 582, 485, 593], "content": ") and a fundamental unit . Therefore if and only", "parent_index": 5, "line_index": 5}, {"bbox": [125, 594, 487, 606], "content": "if , where and where denotes the", "parent_index": 5, "line_index": 6}, {"bbox": [125, 606, 487, 617], "content": "quadratic residue symbol in . Since , we have proved", "parent_index": 5, "line_index": 7}, {"bbox": [126, 617, 487, 630], "content": "that if and only if the prime ideal above splits in the quadratic", "parent_index": 5, "line_index": 8}, {"bbox": [126, 630, 486, 641], "content": "extension . But if we fix and , this happens for exactly half of the values", "parent_index": 5, "line_index": 9}, {"bbox": [125, 641, 297, 654], "content": "of satisfying , .", "parent_index": 5, "line_index": 10}, {"bbox": [136, 652, 487, 666], "content": "If and , then , and we have to choose in such a", "parent_index": 6, "line_index": 0}, {"bbox": [127, 665, 486, 677], "content": "way that is unramified outside . The residue symbols are defined", "parent_index": 6, "line_index": 1}, {"bbox": [125, 677, 486, 689], "content": "as Kronecker symbols via the splitting of in the quadratic extension .", "parent_index": 6, "line_index": 2}, {"bbox": [126, 689, 390, 702], "content": "With these modifactions, the above arguments remain valid.", "parent_index": 6, "line_index": 3}]
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[]
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[{"bbox": [164, 127, 209, 138], "content": "h_{2}(K)=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [232, 127, 278, 138], "content": "h_{2}(K)>2", "parent_index": 0, "subtype": "inline"}, {"bbox": [434, 153, 447, 161], "content": "K_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [471, 150, 484, 161], "content": "\\widetilde{K}_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 164, 148, 173], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [294, 164, 325, 174], "content": "d_{2}\\,=\\,p", "parent_index": 1, "subtype": "inline"}, {"bbox": [350, 164, 382, 173], "content": "d_{3}~=~r", "parent_index": 1, "subtype": "inline"}, {"bbox": [404, 176, 432, 185], "content": "d_{2}=8", "parent_index": 1, "subtype": "inline"}, {"bbox": [446, 177, 482, 185], "content": "d_{3}=-8", "parent_index": 1, "subtype": "inline"}, {"bbox": [181, 191, 186, 198], "content": "p", "parent_index": 1, "subtype": "inline"}, {"bbox": [210, 191, 215, 196], "content": "r", "parent_index": 1, "subtype": "inline"}, {"bbox": [255, 189, 265, 197], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 188, 330, 198], "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [354, 188, 399, 197], "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [427, 189, 433, 196], "content": "h", "parent_index": 1, "subtype": "inline"}, {"bbox": [221, 200, 231, 209], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 199, 323, 210], "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "parent_index": 1, "subtype": "inline"}, {"bbox": [348, 199, 387, 210], "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 215, 132, 219], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [157, 215, 162, 222], "content": "\\rho", "parent_index": 1, "subtype": "inline"}, {"bbox": [433, 225, 442, 232], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [188, 236, 194, 244], "content": "h", "parent_index": 1, "subtype": "inline"}, {"bbox": [256, 235, 306, 245], "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "parent_index": 1, "subtype": "inline"}, {"bbox": [368, 236, 374, 246], "content": "\\mathfrak{p}", "parent_index": 1, "subtype": "inline"}, {"bbox": [478, 239, 485, 244], "content": "\\alpha", "parent_index": 1, "subtype": "inline"}, {"bbox": [378, 248, 398, 257], "content": "4{\\cal O}_{K}", "parent_index": 1, "subtype": "inline"}, {"bbox": [469, 250, 474, 255], "content": "\\mathfrak{a}", "parent_index": 1, "subtype": "inline"}, {"bbox": [187, 259, 234, 270], "content": "[\\varepsilon/{\\mathfrak a}]=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [293, 259, 327, 271], "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "parent_index": 1, "subtype": "inline"}, {"bbox": [363, 259, 386, 270], "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "parent_index": 1, "subtype": "inline"}, {"bbox": [205, 272, 215, 280], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [247, 272, 270, 282], "content": "\\big[\\cdot\\big/\\cdot\\big]", "parent_index": 1, "subtype": "inline"}, {"bbox": [445, 272, 455, 281], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 284, 271, 294], "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "parent_index": 1, "subtype": "inline"}, {"bbox": [147, 295, 195, 306], "content": "[\\pi/\\rho]=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [218, 295, 307, 306], "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "parent_index": 1, "subtype": "inline"}, {"bbox": [159, 309, 228, 320], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [253, 307, 326, 319], "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [400, 307, 454, 319], "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "parent_index": 2, "subtype": "inline"}, {"bbox": [198, 321, 249, 333], "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [286, 321, 368, 333], "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [180, 334, 194, 345], "content": "\\widetilde{K}_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [219, 334, 277, 347], "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [138, 347, 164, 360], "content": "\\widetilde{F}_{1}/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [188, 347, 216, 360], "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [215, 360, 353, 373], "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [200, 375, 251, 386], "content": "[\\pi/\\rho^{\\prime}]=-1", "parent_index": 2, "subtype": "inline"}, {"bbox": [329, 375, 336, 384], "content": "{\\mathfrak{r}}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [385, 374, 410, 386], "content": "\\widetilde{F}_{1}/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [246, 474, 274, 484], "content": "\\widetilde{q}_{2}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [289, 474, 317, 484], "content": "\\widetilde{q}_{2}=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [410, 474, 460, 484], "content": "h_{2}(K_{1})=4", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 485, 139, 496], "content": "\\widetilde{K}_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [402, 484, 465, 497], "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 499, 155, 509], "content": "\\widetilde{q}_{2}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [278, 499, 325, 509], "content": "4\\mid h_{2}(K_{2})", "parent_index": 4, "subtype": "inline"}, {"bbox": [392, 499, 437, 509], "content": "4\\mid h_{2}(K_{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 511, 191, 522], "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [217, 511, 265, 521], "content": "[\\pi/\\rho]=+1", "parent_index": 4, "subtype": "inline"}, {"bbox": [365, 523, 400, 533], "content": "\\mathrm{Cl}_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 535, 157, 545], "content": "h_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [221, 535, 265, 545], "content": "2\\mid h_{2}(L_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [300, 534, 362, 545], "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [182, 547, 195, 556], "content": "K_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [425, 547, 452, 557], "content": "L_{1}/F_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 559, 188, 569], "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [254, 559, 300, 569], "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [369, 559, 430, 569], "content": "(E\\,:\\,H)\\;=\\;1", "parent_index": 5, "subtype": "inline"}, {"bbox": [465, 559, 474, 567], "content": "E", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 583, 184, 591], "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "parent_index": 5, "subtype": "inline"}, {"bbox": [304, 586, 310, 591], "content": "\\varepsilon", "parent_index": 5, "subtype": "inline"}, {"bbox": [368, 583, 430, 593], "content": "(E\\,:\\,H)\\;=\\;1", "parent_index": 5, "subtype": "inline"}, {"bbox": [136, 595, 247, 605], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [284, 595, 348, 605], "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [402, 595, 430, 605], "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 607, 258, 616], "content": "F_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [291, 606, 414, 617], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 618, 195, 629], "content": "4\\mid h_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [333, 619, 345, 628], "content": "\\Re_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [378, 621, 383, 626], "content": "\\mathfrak{r}", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 630, 203, 641], "content": "F_{1}(\\sqrt{\\varepsilon})", "parent_index": 5, "subtype": "inline"}, {"bbox": [267, 633, 272, 640], "content": "p", "parent_index": 5, "subtype": "inline"}, {"bbox": [294, 633, 299, 640], "content": "q", "parent_index": 5, "subtype": "inline"}, {"bbox": [137, 646, 142, 650], "content": "r", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 642, 239, 653], "content": "(p/r)=-1", "parent_index": 5, "subtype": "inline"}, {"bbox": [244, 642, 293, 653], "content": "(q/r)=+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 655, 176, 664], "content": "d_{2}=8", "parent_index": 6, "subtype": "inline"}, {"bbox": [198, 655, 222, 664], "content": "p=2", "parent_index": 6, "subtype": "inline"}, {"bbox": [250, 654, 298, 665], "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [405, 653, 442, 665], "content": "2^{h}=(\\pi)", "parent_index": 6, "subtype": "inline"}, {"bbox": [167, 666, 214, 677], "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [310, 669, 315, 676], "content": "\\mathfrak{p}", "parent_index": 6, "subtype": "inline"}, {"bbox": [413, 666, 435, 677], "content": "\\left[\\alpha/2\\right]", "parent_index": 6, "subtype": "inline"}, {"bbox": [308, 681, 313, 686], "content": "^{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [433, 678, 482, 689], "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "parent_index": 6, "subtype": "inline"}]
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# References
[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $\mathrm{Cl}_{2}(k^{1})$ , J. Number Theory 67 (1997), 229–245.
[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27.
[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337.
[4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48.
[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95.
[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.
[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127.
[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.
[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.
[10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260.
[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18.
[12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.
[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkorperturm, Arch. Math. 34 (1980), 307–312.
(E. Benjamin) Mathematics Department, Unity College $E$ -mail address: [email protected]
(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $E$ -mail address: [email protected]
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<html><body>
<h1 data-bbox="276 113 336 123">References </h1>
<p data-bbox="136 129 487 360">[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $\mathrm{Cl}_{2}(k^{1})$ , J. Number Theory 67 (1997), 229–245.
[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27.
[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337.
[4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48.
[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95.
[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.
[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127.
[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.
[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.
[10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260.
[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18.
[12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.
[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkorperturm, Arch. Math. 34 (1980), 307–312. </p>
<p data-bbox="137 368 363 388">(E. Benjamin) Mathematics Department, Unity College $E$ -mail address: [email protected] </p>
<p data-bbox="138 395 390 415">(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $E$ -mail address: [email protected] </p>
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0003244v1
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[]
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[{"bbox": [152, 142, 180, 151], "content": "\\mathrm{Cl}_{2}(k^{1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 381, 144, 387], "content": "E", "parent_index": 2, "subtype": "inline"}, {"bbox": [138, 408, 145, 414], "content": "E", "parent_index": 3, "subtype": "inline"}]
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[]
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