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281_10	Finances In the financial year ended 31 July 2011, City had a total income (including share of joint ventures) of £178.6 million (2008/09 – £174.4 million) and total expenditure of £183.62 million (2008/09 – £178.82 million). Key sources of income included £39.58 million from Funding Council grants (2008/09 – £39.52 million), £116.91 million from tuition fees and education contracts (2008/09 – £104.39 million), £7.86 million from research grants and contracts (2008/09 – £9.29 million), £1.04 from endowment and investment income (2008/09 – £1.83 million) and £15.05 million from other income (2008/09 – £19.37 million). During the 2010/11 financial year, City had a capital expenditure of £9.77 million (2008/09 – £16.13 million). At year end, City had reserves and endowments of £112.89 million (2009/10 – £110.05 million) and total net assets of £147.64 million (2008/09 – £147.27 million). Academic profile Courses and rankings
281_11	City, University of London, offers Bachelor's, Master's, and Doctoral degrees as well as certificates and diplomas at both undergraduate and postgraduate level. More than two-thirds of City's programmes are recognised by the appropriate professional bodies such as the BCS, BPS, CILIP, ICE, RICS, HPC etc. in recognition of the high standards of relevance to the professions. The university also has an online careers network where over 2,000 former students offer practical help to current students. The City Law School offers courses for undergraduates, postgraduates, master graduates and professional courses leading to qualification as a solicitor or barrister, as well as continuing professional development. Its Legal Practice Course has the highest quality rating from the Solicitors Regulation Authority.
281_12	The Department of Radiography (part of the School of Community and Health Sciences) offers two radiography degrees, the BSc (Hons) Radiography (Diagnostic Imaging) and BSc (Hons) Radiography (Radiotherapy and Oncology), both of which are recognised by the Health Professions Council (HPC). Partnerships and collaborations CETL Queen Mary, University of London, and City, University of London, were jointly awarded Centre for Excellence in Teaching and Learning (CETL) status by the Higher Education Funding Council for England (HEFCE) in recognition of their work in skills training for 3,000 students across six healthcare professions. City of London City, University of London, has links with businesses in the City of London. City has also joined forces with other universities such as Queen Mary and the Institute of Education (both part of the University of London) with which it jointly delivers several leading degree programmes.
281_13	LCACE London Centre for Arts and Cultural Exchange is a consortium of nine universities. It was established in 2004 to foster collaboration and to promote and support the exchange of knowledge between the consortium's partners and London's arts and cultural sectors. The nine institutions involved are: University of the Arts London; Birkbeck, University of London; City, University of London; The Courtauld Institute of Art; Goldsmiths, University of London; Guildhall School of Music & Drama; King's College London; Queen Mary, University of London, and Royal Holloway, University of London.
281_14	WC2 University Network City is a founding member of the WC2 University Network, a network of universities developed with the goal of bringing together leading universities located in the heart of major world cities in order to address cultural, environmental and political issues of common interest to world cities and their universities. In addition to City, University of London, the founding members of WC2 members are: City University of New York, Technische Universität Berlin, Universidade de São Paulo, Hong Kong Polytechnic University, Universidad Autonoma Metropolitana, Saint Petersburg State Polytechnical University, Politecnico di Milano, University of Delhi, Northeastern University Boston and Tongji University.
281_15	Erasmus Mundus MULTI City was selected as the sole British university to take part in the selective Erasmus Mundus MULTI programme, funded by the European Commission to promote scientific exchange between Europe and the industrialised countries of South-East Asia. It is the first Erasmus program to involve universities outside of Europe. In addition to City, the partner universities are: Aix-Marseille University (France), Univerzita Karlova v Praze (Czech Republic), Freie Universität Berlin (Germany), Universität des Saarlandes (Germany), Università di Pisa (Italy), Universidad de Sevilla (Spain), The Hong Kong Polytechnic University (Hong Kong, SAR China), Universiti Brunei Darussalam (Brunei), University of Macau (Macau, SAR China), Nanyang Technological University (Singapore), and National Taiwan University (Taiwan).
281_16	UCL Partners City has joined the executive group of UCL Partners, one of five accredited academic health science groups in the UK. City was invited to join the partnership in recognition of its expertise in nursing, allied health, health services research and evaluation and health management. Student life Students' Union The City Students' Union is run primarily by students through three elected sabbatical officers, an executive committee and a union council, with oversight by a trustee board. The Students' Union provides support, representation, facilities, services, entertainment and activities for its members. It is run for students, by students.
281_17	Student media City currently has two student-run media outlets, including Carrot Radio, which was co-founded by journalism postgraduates Jordan Gass-Poore' and Winston Lo in the autumn of 2018. Carrot Radio currently records weekday podcasts. The second is the student-led online magazine, Carrot Magazine. They recently released their first print magazine in December 2017. Other For a number of years, City students have taken part in the annual Lord Mayor's Show, representing the university in one of the country's largest and liveliest parades. Sustainability ranking City ranked joint 5th out of the 168 universities surveyed in the 2019 People & Planet league table of the most sustainable UK universities having climbed from 7th place in the 2016 league. In both the 2016 and 2019 rankings, it was the highest ranking University of London institution, and one of only four London institutions in the top twenty.
281_18	The league table's Fossil Free Scorecard report, drawn from Freedom of Information requests, found that £800,000 (6.4%) of City's £12.5m endowment was invested in fossil fuels, and that the institution had not made a public commitment to fossil fuel divestment. It also noted nearly £1m of research funding into renewables since 2001 with just £64k of total funding from fossil fuel companies; and no honorary degrees or board positions held by fossil fuel executives. Notable people Notable alumni Government, politics and society Mahatma Gandhi – Leader of the Indian Independence Movement, graduated in 1891 from the Inns of Court School of Law (now part of The City Law School)
281_19	Muhammad Ali Jinnah – founder of Pakistan, first Governor-General of Pakistan graduated from the Inns of Court school of Law (now part of The City Law School) Margaret Thatcher – Conservative Party Prime Minister of the United Kingdom from 1979 to 1990, graduated from the Inns of Court School of Law (now part of The City Law School) Clement Attlee – Labour Prime Minister of the United Kingdom from 1945 to 1951 H. H. Asquith – Liberal Prime Minister of the United Kingdom from 1908 to 1916 Tony Blair – Labour Party Prime Minister of the United Kingdom from 1997 to 2007, graduated from the Inns of Court School of Law (now part of The City Law School) Christos Staikouras – Finance Minister of Greece from 2019 to present Roderic Bowen – Welsh Liberal Party politician Robert Chote – chief of the Office for Budget Responsibility; former director of Institute for Fiscal Studies Ali Dizaei – former police commander
281_20	Jody Dunn – Liberal Democrat politician, and a barrister specialising in family law Sir James Dutton – Royal Marine general and former deputy commander of the International Security Assistance Force Chloë Fox – Australian politician, former Labor MP for the South Australian electoral district of Bright Syed Sharifuddin Pirzada – Noted Pakistani lawyer & Politician. Also served as 5th secretary general of Organisation of Islamic Cooperation. James Hart – Commissioner of the City of London Police David Heath – Politician and Liberal Democrat Member of Parliament for Somerton and Frome Syed Kamall – Conservative Party politician and Member of the European Parliament for the London European Parliament constituency David Lammy – Labour MP for Tottenham Barbara Mensah, judge Liu Mingkang – Chinese Politician and Businessman, current Chairman of the China Banking Regulatory Commission, former Vice-Governor of the China Development Bank
281_21	Jawaharlal Nehru – First Prime Minister of the Republic of India Houda Nonoo – Bahraini Ambassador to the United States Patrick O'Flynn – UK Independence Party MEP Stav Shaffir – Youngest member of the Israeli Knesset, leader of the social justice movement Aris Spiliotopoulos – Minister of Greek Tourism Ivy Williams – First woman to be called to the English bar
281_22	Arts, science and academia
281_23	L. Bruce Archer – British mechanical engineer and Professor of Design Research at the Royal College of Art Susan Bickley – Mezzo-soprano in opera and classical music George Daniels – Horologist, regarded as the greatest watchmaker of modern times and inventor of the coaxial escapement Jerry Fishenden – Technologist, former Microsoft National Technology Officer for the UK Julia Gomelskaya – Ukrainian contemporary music composer, professor of Odessa State Music Academy in Ukraine Norman Gowar – Professor of Mathematics at the Open University and Principal of Royal Holloway College, University of London Michel Guillon - British optometrist and researcher Clare Hammond – Concert pianist David Hirsh – Academic and sociologist Muhammad Iqbal – Muslim poet, philosopher and politician, born in present-day Pakistan, graduated from the Inns of Court School of Law and University of Cambridge
281_24	John Loder – Sound engineer, record producer and founder of Southern Studios, as well as a former member of EXIT Sharon Maguire – Director of Bridget Jones's Diary Rhodri Marsden – Journalist, musician and blogger; columnist for The Independent Robin Milner – Computer scientist and recipient of the 1991 ACM Turing Award Bernard Miles - Actor and founder of the Mermaid Theatre. John Palmer – Instrumental and electroacoustic music composer Sebastian Payne – Journalist Ziauddin Sardar – Academic and scholar of Islamic issues, Commissioner of the Equality and Human Rights Commission Theresa Wallach – Pioneer female engineer, motorcycle adventurer, author, educator and entrepreneur, holder of Brooklands Gold Star. John Hodge – Aeronautical Engineer who played a key role in NASA and America's space race.
281_25	Business and finance
281_26	Winston Set Aung – Politician, Economist and Management Consultant, incumbent Deputy Governor of the Central Bank of Myanmar Brendan Barber – General Secretary of the Trades Union Congress Jonathan Breeze – Founder and CEO of Jet Republic, private jet airline company in Europe Michael Boulos – associate director of Callian Capital Group, and partner of Tiffany Trump William Castell – former Chairman of the Wellcome Trust and a Director of General Electric and BP, former CEO of Amersham plc Peter Cullum – British entrepreneur James J. Greco – former CEO and President of Sbarro Sir Stelios Haji-Ioannou – Founder of easyGroup Tom Ilube CBE, British entrepreneur and Chair of the RFU Bob Kelly – former CEO of Bank of New York Mellon and CFO of Mellon Financial Corporation and Wachovia Corporation Muhtar Kent – former CEO and Chairman of The Coca-Cola Company William Lewis – former CEO Dow Jones Publisher, The Wall Street Journal
281_27	Ian Livingstone – chairman and co-owner, London & Regional Properties Liu Mingkang – former Chairman of the China Banking Regulatory Commission Dick Olver – former Chairman of BAE Systems, member of the board of directors at Reuters Syed Ali Raza – former president and Chairman of the National Bank of Pakistan Martin Wheatley – former CEO of the Financial Conduct Authority Brian Wynter – Governor of the Bank of Jamaica Durmuş Yılmaz – Governor of the Central Bank of Turkey
281_28	Media and entertainment
281_29	Samira Ahmed – Channel 4 News presenter, BBC News presenter, writer and journalist Decca Aitkenhead – journalist Joanna Blythman – Non-fiction writer, Britain's leading investigative food journalist Emily Buchanan – BBC World Affairs correspondent Sally Bundock – BBC Presenter Ellie Crisell – BBC Presenter Imogen Edwards-Jones – Novelist Gamal Fahnbulleh – Sky News Presenter and journalist Mimi Fawaz, BBC presenter and journalist Michael Fish – BBC weatherman Adam Fleming – CBBC Reporter Lourdes Garcia-Navarro – Journalist, Jerusalem foreign correspondent for National Public Radio (NPR) Alex Graham – Chairman of PACT and the Scott Trust Michael Grothaus – Novelist and journalist; author of Epiphany Jones Rachel Horne – BBC and Virgin Radio presenter and journalist Faisal Islam – BBC News Economics Editor Gillian Joseph – Sky News Presenter Kirsty Lang – BBC Presenter and journalist Ellie Levenson – Freelance Journalist and Author
281_30	William Lewis – Journalist and editor of The Daily Telegraph Donal MacIntyre – Investigative journalist Sharon Maguire – Writer and Director, directed Bridget Jones's Diary Rhodri Marsden – Journalist, musician and blogger; columnist for The Independent Sharon Mascall – Journalist, broadcaster and writer; lecturer at the University of South Australia Lucrezia Millarini – Freelance Journalist and ITV Newsreader Dermot Murnaghan – Presenter on Sky News Tiff Needell – Grand Prix driver, Presenter of Fifth Gear on Five Maryam Nemazee – Presenter for Al Jazeera London Linda Papadopoulos – Psychologist, appearing occasionally on TV Catherine Pepinster – journalist, religion writer Raj Persaud – British consultant psychiatrist, broadcaster, and author on psychiatry Richard Preston – Novelist Gavin Ramjaun – Television presenter and journalist Sophie Raworth – Newsreader, presenter on BBC One O'Clock News Apsara Reddy – journalist
281_31	Joel Rubin – World-renowned klezmer clarinetist Ian Saville – British magician Barbara Serra – Presenter for Al Jazeera London Sarah Walker – BBC Radio 3 presenter Josh Widdicombe - Comedian and presenter Mark Worthington – BBC Correspondent
281_32	Notable faculty and staff Rosemary Crompton – Professor of Sociology Roy Greenslade – Journalist Steven Haberman – Professor of Actuarial Science at City, University of London Corinna Hawkes – Professor of Food Policy Rosemary Hollis – Professor of International Politics at City, University of London Jamal Nazrul Islam – Physicist, Mathematician, Cosmologist, Astronomer Ernest Krausz (1931-2018) - Israeli professor of sociology and President at Bar Ilan University David Leigh – Journalist David Marks – Psychologist Penny Marshall – Journalist Stewart Purvis – Broadcaster Denis Smalley – Composer Bill Thompson – Journalist David Willets – Conservative Member of Parliament for Havant; Shadow Secretary of State for Education and Skills
281_33	Vice-Chancellors (Pre-2016)/Presidents (Post-2016) 1966–1974: Sir James Sharp Tait 1974–1978: Sir Edward W. Parkes 1978–1998: Raoul Franklin 1998–2007: David William Rhind 2007–2009: Malcolm Gillies 2009–2010: Julius Weinberg (acting) 2010–2021: Sir Paul Curran 2021–Present: Sir Anthony Finkelstein In popular culture City University's Bastwick Street Halls of Residence in Islington was the first home of MasterChef following its 2005 revival. References External links City, University of London City, University of London, Students Union Lists of Northampton Polytechnic Institute students List of Northampton Polytechnic Institute military personnel, 1914–1918 Optometry schools Schools of informatics Educational institutions established in 1894 1894 establishments in England Venues of the 1908 Summer Olympics Olympic boxing venues Universities UK
282_0	In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order which are all isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties
282_1	A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order exists if and only if is a prime power (where is a prime number and is a positive integer). In a field of order , adding copies of any element always results in zero; that is, the characteristic of the field is . If , all fields of order are isomorphic (see below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted , or , where the letters GF stand for "Galois field".
282_2	In a finite field of order , the polynomial has all elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.) The simplest examples of finite fields are the fields of prime order: for each prime number , the prime field of order , , may be constructed as the integers modulo , . The elements of the prime field of order may be represented by integers in the range . The sum, the difference and the product are the remainder of the division by of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see ).
282_3	Let be a finite field. For any element in and any integer , denote by the sum of copies of . The least positive such that is the characteristic of the field. This allows defining a multiplication of an element of by an element of by choosing an integer representative for . This multiplication makes into a -vector space. It follows that the number of elements of is for some integer . The identity (sometimes called the freshman's dream) is true in a field of characteristic . This follows from the binomial theorem, as each binomial coefficient of the expansion of , except the first and the last, is a multiple of . By Fermat's little theorem, if is a prime number and is in the field then . This implies the equality for polynomials over . More generally, every element in satisfies the polynomial equation .
282_4	Any finite field extension of a finite field is separable and simple. That is, if is a finite field and is a subfield of , then is obtained from by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect. A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field. Existence and uniqueness Let be a prime power, and be the splitting field of the polynomial
282_5	over the prime field . This means that is a finite field of lowest order, in which has distinct roots (the formal derivative of is , implying that , which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of are roots of , as well as the multiplicative inverse of a root of . In other words, the roots of form a field of order , which is equal to by the minimality of the splitting field. The uniqueness up to isomorphism of splitting fields implies thus that all fields of order are isomorphic. Also, if a field has a field of order as a subfield, its elements are the roots of , and cannot contain another subfield of order . In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:
282_6	The order of a finite field is a prime power. For every prime power there are fields of order , and they are all isomorphic. In these fields, every element satisfies and the polynomial factors as It follows that contains a subfield isomorphic to if and only if is a divisor of ; in that case, this subfield is unique. In fact, the polynomial divides if and only if is a divisor of . Explicit construction Non-prime fields Given a prime power with prime and , the field may be explicitly constructed in the following way. One first chooses an irreducible polynomial in of degree (such an irreducible polynomial always exists). Then the quotient ring of the polynomial ring by the ideal generated by is a field of order .
282_7	More explicitly, the elements of are the polynomials over whose degree is strictly less than . The addition and the subtraction are those of polynomials over . The product of two elements is the remainder of the Euclidean division by of the product in . The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions. Except in the construction of , there are several possible choices for , which produce isomorphic results. To simplify the Euclidean division, one commonly chooses for a polynomial of the form
282_8	which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic , irreducible polynomials of the form may not exist. In characteristic , if the polynomial is reducible, it is recommended to choose with the lowest possible that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" , as polynomials of degree greater than , with an even number of terms, are never irreducible in characteristic , having as a root. A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields. In the next sections, we will show how the general construction method outlined above works for small finite fields.
282_9	Field with four elements The smallest non-prime field is the field with four elements, which is commonly denoted or It consists of the four elements such that and for every the other operation results being easily deduced from the distributive law. See below for the complete operation tables. This may be deduced as follows from the results of the preceding section. Over , there is only one irreducible polynomial of degree : Therefore, for the construction of the preceding section must involve this polynomial, and Let denote a root of this polynomial in . This implies that and that and are the elements of that are not in . The tables of the operations in result from this, and are as follows:
282_10	A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of by , the values of must be read in the left column, and the values of in the top row. (Because for every in every ring the division by 0 has to remain undefined.) The map is the non-trivial field automorphism, called Frobenius automorphism, which sends into the second root of the above mentioned irreducible polynomial GF(p2) for an odd prime p For applying the above general construction of finite fields in the case of , one has to find an irreducible polynomial of degree 2. For , this has been done in the preceding section. If is an odd prime, there are always irreducible polynomials of the form , with in .
282_11	More precisely, the polynomial is irreducible over if and only if is a quadratic non-residue modulo (this is almost the definition of a quadratic non-residue). There are quadratic non-residues modulo . For example, is a quadratic non-residue for , and is a quadratic non-residue for . If , that is , one may choose as a quadratic non-residue, which allows us to have a very simple irreducible polynomial . Having chosen a quadratic non-residue , let be a symbolic square root of , that is a symbol which has the property , in the same way as the complex number is a symbolic square root of . Then, the elements of are all the linear expressions with and in . The operations on are defined as follows (the operations between elements of represented by Latin letters are the operations in ): GF(8) and GF(27) The polynomial
282_12	is irreducible over and , that is, it is irreducible modulo and (to show this, it suffices to show that it has no root in nor in ). It follows that the elements of and may be represented by expressions where are elements of or (respectively), and is a symbol such that The addition, additive inverse and multiplication on and may thus be defined as follows; in following formulas, the operations between elements of or , represented by Latin letters, are the operations in or , respectively: GF(16) The polynomial is irreducible over , that is, it is irreducible modulo . It follows that the elements of may be represented by expressions where are either or (elements of ), and is a symbol such that
282_13	(that is, is defined as a root of the given irreducible polynomial). As the characteristic of is , each element is its additive inverse in . The addition and multiplication on may be defined as follows; in following formulas, the operations between elements of , represented by Latin letters are the operations in . The field has eight primitive elements (the elements that have all nonzero elements of as integer powers). These elements are the four roots of and their multiplicative inverses. In particular, is a primitive element, and the primitive elements are with less than and coprime with 15 (that is, 1, 2, 4, 7, 8, 11, 13, 14). Multiplicative structure
282_14	The set of non-zero elements in is an abelian group under the multiplication, of order . By Lagrange's theorem, there exists a divisor of such that for every non-zero in . As the equation has at most solutions in any field, is the lowest possible value for . The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. In summary: The multiplicative group of the non-zero elements in is cyclic, and there exists an element , such that the non-zero elements of are . Such an element is called a primitive element. Unless , the primitive element is not unique. The number of primitive elements is where is Euler's totient function. The result above implies that for every in . The particular case where is prime is Fermat's little theorem. Discrete logarithm
282_15	If is a primitive element in , then for any non-zero element in , there is a unique integer with such that . This integer is called the discrete logarithm of to the base . While can be computed very quickly, for example using exponentiation by squaring, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols, see Discrete logarithm for details. When the nonzero elements of are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo . However, addition amounts to computing the discrete logarithm of . The identity allows one to solve this problem by constructing the table of the discrete logarithms of , called Zech's logarithms, for (it is convenient to define the discrete logarithm of zero as being ).
282_16	Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field. Roots of unity Every nonzero element of a finite field is a root of unity, as for every nonzero element of . If is a positive integer, an th primitive root of unity is a solution of the equation that is not a solution of the equation for any positive integer . If is a th primitive root of unity in a field , then contains all the roots of unity, which are . The field contains a th primitive root of unity if and only if is a divisor of ; if is a divisor of , then the number of primitive th roots of unity in is (Euler's totient function). The number of th roots of unity in is .
282_17	In a field of characteristic , every th root of unity is also a th root of unity. It follows that primitive th roots of unity never exist in a field of characteristic . On the other hand, if is coprime to , the roots of the th cyclotomic polynomial are distinct in every field of characteristic , as this polynomial is a divisor of , whose discriminant is nonzero modulo . It follows that the th cyclotomic polynomial factors over into distinct irreducible polynomials that have all the same degree, say , and that is the smallest field of characteristic that contains the th primitive roots of unity. Example: GF(64) The field has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree over ) are primitive elements; and the primitive elements are not all conjugate under the Galois group.
282_18	The order of this field being , and the divisors of being , the subfields of are , , , and itself. As and are coprime, the intersection of and in is the prime field . The union of and has thus elements. The remaining elements of generate in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree over . This implies that, over , there are exactly irreducible monic polynomials of degree . This may be verified by factoring over . The elements of are primitive th roots of unity for some dividing . As the 3rd and the 7th roots of unity belong to and , respectively, the generators are primitive th roots of unity for some in . Euler's totient function shows that there are primitive th roots of unity, primitive st roots of unity, and primitive rd roots of unity. Summing these numbers, one finds again elements.
282_19	By factoring the cyclotomic polynomials over , one finds that: The six primitive th roots of unity are roots of and are all conjugate under the action of the Galois group. The twelve primitive st roots of unity are roots of They form two orbits under the action of the Galois group. As the two factors are reciprocal to each other, a root and its (multiplicative) inverse do not belong to the same orbit. The primitive elements of are the roots of They split into six orbits of six elements each under the action of the Galois group. This shows that the best choice to construct is to define it as . In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division. Frobenius automorphism and Galois theory In this section, is a prime number, and is a power of . In , the identity implies that the map
282_20	is a -linear endomorphism and a field automorphism of , which fixes every element of the subfield . It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. Denoting by the composition of with itself times, we have It has been shown in the preceding section that is the identity. For , the automorphism is not the identity, as, otherwise, the polynomial would have more than roots. There are no other -automorphisms of . In other words, has exactly -automorphisms, which are In terms of Galois theory, this means that is a Galois extension of , which has a cyclic Galois group. The fact that the Frobenius map is surjective implies that every finite field is perfect. Polynomial factorization If is a finite field, a non-constant monic polynomial with coefficients in is irreducible over , if it is not the product of two non-constant monic polynomials, with coefficients in .
282_21	As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, every computer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields. Irreducible polynomials of a given degree The polynomial factors into linear factors over a field of order . More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order .
282_22	This implies that, if then is the product of all monic irreducible polynomials over , whose degree divides . In fact, if is an irreducible factor over of , its degree divides , as its splitting field is contained in . Conversely, if is an irreducible monic polynomial over of degree dividing , it defines a field extension of degree , which is contained in , and all roots of belong to , and are roots of ; thus divides . As does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it. This property is used to compute the product of the irreducible factors of each degree of polynomials over ; see Distinct degree factorization. Number of monic irreducible polynomials of a given degree over a finite field The number of monic irreducible polynomials of degree over is given by where is the Möbius function. This formula is almost a direct consequence of above property of .
282_23	By the above formula, the number of irreducible (not necessarily monic) polynomials of degree over is . A (slightly simpler) lower bound for is One may easily deduce that, for every and every , there is at least one irreducible polynomial of degree over . This lower bound is sharp for . Applications In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis of several widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field. In coding theory, many codes are constructed as subspaces of vector spaces over finite fields.
282_24	Finite fields are used by many error correction codes, such as Reed–Solomon error correction code or BCH code. The finite field almost always has characteristic of 2, since computer data is stored in binary. For example, a byte of data can be interpreted as an element of . One exception is PDF417 bar code, which is . Some CPUs have special instructions that can be useful for finite fields of characteristic 2, generally variations of carry-less product. Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing them modulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.
282_25	Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods. Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields. The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates. Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices. In arithmetic combinatorics finite fields and finite field models are used extensively, such as in Szemerédi's theorem on arithmetic progressions. Extensions
282_26	Algebraic closure A finite field is not algebraically closed: the polynomial has no roots in , since for all in . Fix an algebraic closure of . The map sending each to is called the th power Frobenius automorphism. The subfield of fixed by the th iterate of is the set of zeros of the polynomial , which has distinct roots since its derivative in is , which is never zero. Therefore that subfield has elements, so it is the unique copy of in . Every finite extension of in is this for some , so The absolute Galois group of is the profinite group
282_27	Like any infinite Galois group, may be equipped with the Krull topology, and then the isomorphisms just given are isomorphisms of topological groups. The image of in the group is the generator , so corresponds to . It follows that has infinite order and generates a dense subgroup of , not the whole group, because the element has infinite order and generates the dense subgroup One says that is a topological generator of . Quasi-algebraic closure Although finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of Artin and Dickson proved by Chevalley (see Chevalley–Warning theorem).
282_28	Wedderburn's little theorem A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, and hence are finite fields. This result holds even if we relax the associativity axiom to alternativity, that is, all finite alternative division rings are finite fields, by the Artin–Zorn theorem. See also Quasi-finite field Field with one element Finite field arithmetic Finite ring Finite group Elementary abelian group Hamming space Notes References W. H. Bussey (1905) "Galois field tables for pn ≤ 169", Bulletin of the American Mathematical Society 12(1): 22–38, W. H. Bussey (1910) "Tables of Galois fields of order < 1000", Bulletin of the American Mathematical Society 16(4): 188–206, External links Finite Fields at Wolfram research.
283_0	Jack Spigot (4 May 1818 –June 1843) was a British Thoroughbred racehorse that won the 1821 St. Leger Stakes and was a sire in the early 19th century. His paternity is attributed to either Ardrossan or Marmion. His mother was a blind mare with a difficult temperament, whose unpredictable behavior necessitated that he be raised by a foster mare. He was named after one of his owner's tenant farmers, Jack Faucet. He won four of his six career starts before being retired from racing in early 1823. He is not considered to be a good sire. Jack Spigot died in June 1843 and was buried at Bolton Hall. Background Jack Spigot was foaled on 4 May 1818 on Middleham Moor near Bolton Hall in Leyburn, the family seat of his breeder Thomas Orde-Powlett, who was a younger brother of the Baron Bolton. Thomas Orde-Powlett was also a cousin of William Orde, Jr. the owner of Beeswing.
283_1	Jack Spigot's dam, a sister to the stallion Bourbon, was bred by two stallions in the year preceding Jack Spigot's birth. She was first bred to Marmion on 14 April and then by Ardrossan in June 1817. Given the timing of Jack Spigot's birth, Ardrossan is likely his sire. Ardrossan was an unbeaten racehorse in three starts and stood at Rushyford near Durham. Marmion was only defeated once in eight career starts and is an ancestor of the influential broodmare Pocahontas.
283_2	Jack Spigot's dam had a fractious temperament and had been blind and barren for four years before Jack Spigot's birth. Her first foal died at a young age in 1814 and she was owned by H. Peirse until she was sold to Powlett for a small sum. While owned by Peirse, the mare had killed other horses by kicking, notably Reveller's dam Rosette in 1816. She kicked about so much in the foaling paddock after Jack Spigot's birth that in order to prevent the death of her second foal, he had to be raised by a surrogate. A dapple grey foster mare similar in colouring to his own dam was procured from one of Powlett's tenants, Jack Faucet, to raise the foal. The sister to Bourbon mare also produced the grey colt Isaac in 1831 (sired by Figaro), who was considered to be the "best Cup horse of his day", running in flat and steeplechase races until he was 15 years old winning 53 races out of 172 starts.
283_3	Before the horse's racing career, Powlett suggested that the colt be named "Jack Faucet" in honour of John Faucet. Faucet objected to the choice on the grounds that the horse was a good candidate for winning that year's St. Leger. Powlett allegedly quipped, "Well John, a Faucet's nothing without a Spigot" and the colt was subsequently named "Jack Spigot." Jack Spigot was the first horse that John F. Herring painted from life and the artist painted several portraits of the racehorse during and after his racing career. Racing career Jack Spigot was trained at Middleham by Isaac Blades ("J. Blades") who worked exclusively for Powlett. The colt was ridden in his early engagements by Bill Scott. After his St. Leger win, Jack Spigot developed an intense dislike of the jockey and would react violently if Scott attempted to approach the horse or even spoke in his presence. Jack Spigot was not raced extensively, starting only six times and winning four races including the St Leger.
283_4	1820: two-year-old season In his only start as a two-year-old, Jack Spigot, ridden by Bill Scott, won a 320 guinea sweepstakes race at Doncaster Racecourse beating Mr. Riddell's colt Colwell. 1821: three-year-old season Jack Spigot did not run until the autumn Doncaster meeting and on 17 September started in the St. Leger Stakes against 12 other horses including the 1821 Derby winner Gustavus. Coronation was the front runner until three quarters of a mile when Lunatic took the lead for another half-mile. Jack Spigot and the mare Fortuna overtook Coronation with Jack Spigot edging out Fortuna at the finish by a margin of half a length. Two days later he won the Foal Stakes over a distance of one and a half miles beating his only competitor, the filly My Lady, in a "very excellent race."
283_5	1822–1823: four and five-year-old seasons Returning as a four-year-old and refusing to be ridden by Bill Scott, Jack Spigot won the 450-guinea Newcastle Convivial Stakes ridden by Robert Johnson. In his only other start of 1822, he was third and last in the 3.25-mile Preston Cup, losing to Reveller and the 11-year-old Dr. Syntax. Jack Spigot only ran once more, on 14 April 1823 he was unplaced in the Craven Stakes won by the Duke of Rutland's colt Scarborough. Thomas Orde-Powlett retired from racing in September 1823 and put his horses up for sale at the Doncaster meeting. Jack Spigot was retained as a breeding stallion at Bolton Hall.
283_6	Stud career Jack Spigot was retired to stud duty at Bolton Hall in 1824. He is not considered to be a good sire. Jack Spigot died in June 1843 and was buried in Yew Tree Court at Bolton Hall, his grave at one time surrounded by eight yew trees. Some of his bones were exhumed when a water pipe was laid across his grave years after his death. A cannon bone was recovered, set in silver and used to make a letter weight for Lord Bolton. Pedigree References 1818 racehorse births 1843 racehorse deaths Racehorses bred in the United Kingdom Racehorses trained in the United Kingdom Thoroughbred family 5-a St Leger winners
284_0	"Jesus of Suburbia" is a song by American rock band Green Day. It was released as the fifth and final single from the group's seventh studio album, American Idiot, and the second song on the album. With the song running for 9 minutes and 8 seconds, it is Green Day's second longest song (with the band's longest song being fellow American Idiot song "Homecoming", which runs for 9 minutes and 18 seconds) and the group's longest song to be released as a single. The studio version of the song was considered to be unfriendly for radio, so it was cut down to 6½ minutes for the radio edit. The long version was still played on many album rock and alternative rock radio stations. At most live shows on the first leg of the group's 21st Century Breakdown World Tour, the band would pick a member from the audience to play guitar for the song. The single has sold 205,000 copies as of July 2010. Despite its commercial success, the song is the only hit single from the American Idiot album not to be
284_1	included on the band's greatest hits album God's Favorite Band.
284_2	Background American Idiot is a concept album that describes the story of a central character named Jesus of Suburbia, an anti-hero created by Billie Joe Armstrong. It is written from the perspective of a lower-middle-class suburban American teen, raised on a diet of "soda pop and Ritalin." Jesus hates his town and those close to him, so he leaves for The City. "Jesus of Suburbia" was the second multi-part song the group formed. Armstrong said it took "a long time" to write the song. Dirnt said that it came about from natural rehearsing between the trio. The song was an extension of Armstrong's desire to write the "Bohemian Rhapsody" of the future.
284_3	As the song changes into different sections, Armstrong’s guitars were recorded differently. The musicians would "split the signal from the guitar and send it into an amp while simultaneously going direct with it," to achieve a sound reminiscent of "Revolution" by the Beatles or the style of David Bowie guitarist Mick Ronson. In addition, an overdrive pedal was employed to accentuate gain from the instrument, producing a "punchy" sound to each chord. For the first two sections of the song, Cool emulated Ginger Baker and Charlie Watts, two English drummers from the 1960s. For the final three, he drums in his style: "I'm tipping my hat to all these great drummers that I love, and then I kick the door down and do it … my style." In addition to Watts, Cool pulled inspiration from Keith Moon and Alex Van Halen. The song was composed by Green Day (with Billie Joe Armstrong writing the lyrics), and was co-produced by Rob Cavallo. "Jesus of Suburbia" has five movements:
284_4	I. "Jesus of Suburbia" (0:00 – 1:51) II. "City of the Damned" (1:51 – 3:42) III. "I Don't Care" (3:42 – 5:25) IV. "Dearly Beloved" (5:25 – 6:30) V. "Tales of Another Broken Home" (6:30 – 9:08)
284_5	Music videos Two versions of the "Jesus of Suburbia" music video exist, directed by Samuel Bayer (who also directed the music videos for the first four singles released from the American Idiot album). The official music video premiered on October 14, 2005 in the UK and on October 25, 2005 on the MTV network for viewers in the US. One version is a 12-minute edit, complete with a plot and dialogue; the other is a six and a half-minute director's cut, inclusive solely of the music itself and devoid of additives. The twelve-minute version is censored, whereas the six-minute version is not. The video starred Lou Taylor Pucci as Jesus. Jesus' love interest (Whatsername) was played by Kelli Garner. Jesus' mother was portrayed by Canadian actress Deborah Kara Unger. Although Armstrong was originally tipped to provide the acting role of the main character, this was altered during pre-filming.
284_6	The plot of the video essentially follows that of the song. Despite the fact it is the second track, the video reveals Jesus' and Whatsername's relationship before it is revealed in the story. The video pays homage to "1979" by Smashing Pumpkins—it also made use of the snorricam which created the videos' notable up close shots in the convenience store and party scenes. Live performances It has been played at most of the group's concerts since its release. At many concerts on the 21st Century Breakdown World Tour the band picked an audience member from the crowd to play guitar to the song. The song holds the record of the longest performance on the UK television programme, Top of the Pops at 9 minutes and 10 seconds on November 6, 2005.
284_7	Critical reception
284_8	Since its release, "Jesus of Suburbia" has received universal critical acclaim. People magazine called the song "epic" and a "magnificent nine-minute rock opera." It is often recognized as one of Green Day's greatest songs. It was voted the greatest Green Day song of all time in a Rolling Stone readers poll in September 2012. Magnet considered the song underrated, saying "Some will look at this choice and sniff, “How the hell can you call that underrated?” ... But how in the world can you call it “overrated” when the five-movement, nine-plus-minute song bobs and weaves its way through standard-issue pop punk (“Jesus Of Suburbia”), a piano-laced interlude (“City Of The Damned”), the slobbering, thundering middle section (“I Don’t Care”), acoustic mid-tempo connective tissue (“Dearly Beloved”) and an outsized, anthemic curtain call (the spectacularly good “Tales Of Another Broken Home”), all in service of a tale of bored rebellion as nuanced as Pete Townshend’s Quadrophenia and as
284_9	powerful as any of Paul Westerberg’s snot-nosed teenage character studies?".
284_10	Credits and personnel Songwriting – Billie Joe Armstrong, Mike Dirnt, Tré Cool Production – Rob Cavallo, Green Day Track listings 10" Charts Certifications References External links Green Day plan ambitious video for next single Think Green Day's "September" Clip Is Epic? Just Wait For "Jesus Of Suburbia" 2003 songs 2004 songs 2005 singles Green Day songs Songs written by Billie Joe Armstrong Music videos directed by Samuel Bayer American Idiot Song recordings produced by Rob Cavallo Music medleys Songs written by Mike Dirnt Songs written by Tré Cool
285_0	Escape from Rubbish Island is the fifth studio album by English rock band the Wonder Stuff. It was released on 27 September 2004, through the IRL record label. The band had broken up in 1994, had reformed in 2000 and had been playing shows sporadically over the next few years. Frontman Miles Hunt began making drum loops in his home studio, and his flatmate, former Radical Dance Faction member Mark McCarthy, added bass over them. An argument between Hunt and bandmate Martin Gilks resulted in the latter leaving, followed by violinist Martin Bell soon after. Hunt continued working on the tracks, recording at Vada Studios in 2004 with Matt Terry producing. The album saw a return to the band's rock sound of their early albums.
285_1	Escape from Rubbish Island received generally favourable reviews from critics, some of whom commented on Hunt's lyrics. "Better Get Ready for a Fist Fight" reached number 95 in the UK Singles Chart. "Better Get Ready for a Fist Fight" was released as the lead single in January 2005, followed the next month by a joint release of "Bile Chant" and "Escape from Rubbish Island". To promote the album, the band embarked on tours of the United Kingdom and the United States. The US release of the album, which coincided with the tour in that territory, featured alternate mixes and additional guitar parts.
285_2	Background and production From 1988 to 1993, the Wonder Stuff released four studio albums; the band then broke up in June 1994. Frontman Miles Hunt performed as a solo artist briefly, before forming Vent 414; the other members of the band, guitarist Malcolm Treece, bassist Paul Clifford, and drummer Martin Gilks, formed Weknowwhereyoulive. The Wonder Stuff, with the addition of former violinist / banjo player Martin Bell, and new members Stuart Quinell and Pete Whittaker, reunited for a show in 2000, initially as a one-off. Due to demand, the single gig was expanded to five, and the band continued to tour infrequently over the next few years. Hunt wrote new songs in a home studio that he had built in London with Radical Dance Faction member Mark McCarthy, who played bass, in 2003.
285_3	The pair had been sharing a flat with a friend, and Hunt had bought a computer and recording software. In an attempt to learn the software, Hunt looped drum beats, and he asked McCarthy to play over them. Inspired by McCarthy's playing ability, Hunt began adding guitar parts. Over the next two months, Hunt saw an album's worth of material forming; he was aware that the music would be different from that on his two solo releases, Hairy on the Inside (1999) and The Miles Hunt Club (2002). Around this time, the Wonder Stuff's original manager, Les Johnson, introduced Hunt to Matt Terry, a producer who owned his own studio in Stratford-upon-Avon. Terry was friends with Johnson's son, Luke, a drummer in Amen; Hunt had also known Luke since the latter was a young child. Johnson, who lived in California, was visiting his parents; Hunt said he would pay for Johnson's flight back to the US if he could delay it by a week and help record drums for him, which Johnson agreed to.
285_4	While this was occurring, Hunt and McCarthy were making frequent visits to Stratford, where Hunt's manager, David Jaymes, was forming his own label, IRL. Hunt played them early versions of songs he was working on, which the label was ecstatic about releasing. By December 2003, Gilks left the Wonder Stuff after an argument between Hunt and Gilks; soon afterwards, Bell left as well. Hunt didn't take Bell and Gilks's leaving seriously, as the pair had threatened to leave on prior occasions, and it wasn't until early 2004, when Gilks asked Hunt to remove his gear from a lock-up, that Hunt understood that Bell and Gilks were not expecting to return. Despite being short a few band members, Hunt focused on finishing the album he had been working on. Recording was held at Vada Studios in 2004, with Terry as the producer, and James Edwards as engineer. Paul Tipler mixed the recordings at Gravity Shack Studios in London, before the album was mastered by Kevin Grainger at Wired Masters.
285_5	Composition Escape from Rubbish Island was a return to the straightforward rock sound of the band's earlier albums, especially their debut album, The Eight Legged Groove Machine (1988). Hunt said that the title was "a statement about the way Britain's gone over the last ten years. Politically, socially, musically, it's just very backward looking." In contrast with their third studio album, Construction for the Modern Idiot (1993), which was about growing up, Escape from Rubbish Island tackled escapism and divorce. Edwards supplied additional guitars, while Terry provided additional backing vocals; Hunt's uncle, Bill Hunt, contributed on the organ.
285_6	Hunt wrote the majority of the tracks, except for "Bile Chant" (written by Hunt, McCarthy, and Republica member Jonny Male), "Better Get Ready for a Fist Fight" (written by Hunt, McCarthy, and Male), "Another Comic Tragedy" (written by Hunt and Male), and "Head Count" (written by Hunt and McCarthy). The opening track, "Escape from Rubbish Island", lambasts modern England before becoming introspective, and is followed by "Bile Chant", which features flamenco guitar. "Better Get Ready for a Fist Fight" recalled the sound of Construction for the Modern Idiot, and is followed by "Another Tragic Comedy", which tackles the topic of relationships. "Head Count" is a goth-esque track that incorporates an organ; "One Step at a Time" contains elements of funk. The closing track, "Love's Ltd", has Celtic flourishes, with whistling by Geoffrey Kelly.
285_7	Release A financial backer of IRL proposed to Hunt that he release the album under the Wonder Stuff name. After realising that he had put into the album the same effort that he had with the previous Wonder Stuff albums, he decided to put it out under the name. Hunt invited Treece to join him and McCarthy on tour; Johnson was unable to secure a work permit, and was only able to play a few dates before being replaced by former Love in Reverse member Andres Karu, who had drummed for Hunt previously on The Miles Hunt Club. According to Hunt, the ex-members took over the band's website, criticizing the new line-up as "nothing but Miles Hunt and a bunch of his mates going out playing Wonder Stuff songs. To which," Hunt continued, "I could only ask 'isn't that what it always had been? Escape from Rubbish Island was released on 27 September 2004, through IRL. The band toured the United Kingdom until mid-October.
285_8	"Better Get Ready for a Fist Fight" was released as a single on 11 October 2004, with "Apple of My Eye" and "Safety Pin Stuck in My Heart" as extra tracks. "Bile Chant" and "Escape from Rubbish Island" were released as a joint single on 21 February 2005, with remixes of both songs as extra tracks. On 1 March 2005, a music video for "Escape from Rubbish Island" was posted online. The band was invited to tour the United States by touring agent Marc Geiger, who ran a label that was interested in releasing the album there. Escape from Rubbish Island was released in the US in March 2005; as Hunt was unhappy with some of the original mixes, he altered a few of them and had Treece add new guitar parts to some of the tracks. Later in the month, the band played a handful of UK shows. In April and May 2005, the band embarked on a tour of the US, with As Fast As. Reception
285_9	Escape from Rubbish Island was met with generally positive reviews from music critics. AllMusic reviewer John D. Luerssen wrote that the album "may not match" the quality of their third studio album, Never Loved Elvis (1991), but "it boasts some superb songs in the band's unique indie folk/rock style heightened by Hunt's sorely-needed, wry observations." He added that the album "may be littered with a couple of disposable songs... but with irresistibly melodic, attitudinal numbers like 'Back to Work' and 'Another Comic Tragedy', the Wonder Stuff still manage to say it all with their moniker."
285_10	Stylus Magazine Bjorn Randolph highlighted Hunt's lyrics: "He's clearly got a hard case of the older/wisers, and the gleeful misanthropy of the classic Stuffies has been replaced with a wistful air, filled with regrets, coulda-beens and shoulda-beens." Patrick Schabe of PopMatters found Hunt's lyrics to be "as wry and bitter and sneering as ever" but found the music to be "missing a piece of the formula that made up the old, familiar Wonder Stuff." Chart Attack writer David Missio said that a few people "will enjoy Escape From Rubbish Island's Bon Jovi/John Mellencamp sound", though the "brash lyrics" make it fall "much too flat to be a successful comeback album". "Better Get Ready for a Fist Fight" reached number 95 on the UK Singles Chart. Track listing All songs written by Miles Hunt, except where noted. Personnel Personnel per booklet.
285_11	The Wonder Stuff Miles Hunt – lead vocals, guitars, keyboards, programming, percussion Malcom Treece – guitar, backing vocals Mark McCarthy – bass Additional musicians Luke Johnson – drums James Edwards – additional guitars Matt Terry – additional backing vocals Geoffrey Kelly – whistles, flute Bill Hunt – organ Production Matt Terry – producer James Edwards – engineer Paul Tipler – mixing Kevin Grainger – mastering Design Tony Bartolo – photography Miles Hunt – photography Mark McCarthy – photography Alan Robertson – sleeve design References External links Escape from Rubbish Island at YouTube (streamed copy where licensed) 2004 albums The Wonder Stuff albums
286_0	The Mahimal (), also known as Maimal (), are a Bengali Muslim community of inland fishermen predominantly indigenous to the Sylhet Division of Bangladesh and the Barak Valley in Assam, India. Origins According to the traditions of the community, the word Mahimal comes from the Persian word māhi (ماهی) meaning fish and the Arabic word mallāḥ (ملاح) meaning boatman. The Mahimal are said to become Muslims through the efforts of the Sufi saint, Shah Jalal, and his disciples. They are found along the banks of the Sonai and Barak rivers, predominantly in Assam's Barak Valley districts though some can also be found in the Sylhet District. The community converse in the Sylheti dialect of the Bengali language.
286_1	Present circumstances The Mahimal were a community of inland fishermen, but most are now settled agriculturists. They are mainly marginal farmers, growing paddy and vegetables. A small number of Mahimal have taken petty trade. The Mahimal live in multi-ethnic villages, occupying their own quarters, referred to as paras. They are strictly endogamous and marry close kin. Historically, the community practised village exogamy, but this is no longer the case.
286_2	Traditionally, the Mahimals are localised on the banks and nearby areas of rivers and other natural water bodies owing to their customary occupation of fishing. So, roads and other means of modern communications lack in their villages. Even there are some village like Kalachori Par where water remains at least for 6-months. The flood damages all paddy fields; there is no communication system, no road, no electricity, and the percentage of literacy is 1%. The government is not taking any steps and the gram panchayat are considered corrupt, taking all the money with the community going backward day by day. On the socio-economic front also, they are lagging behind the other communities due to their illiteracy and backwardness in education. Due to all-round backwardness, they have been the easy prey in the clutches of the so-called high caste people.
286_3	History Visualising an abundance of opportunities, two Sardars of the Mahimal community, Raghai and Basai, led the community to migrate to Panchakhanda (present-day Beanibazar). The migration was the aftermath of the developmental tasks undertaken by Kalidas Pal, the erstwhile Hindu zamindar of Panchakhanda. The Mahimals subsequently maintained a presence in Beanibazar into modern times.
286_4	In 1913, Mahimals helped in the development of Sylhet Government Alia Madrasah by raising funds following a request by the Education Minister of Assam, Syed Abdul Majid, to the Muslim Fisherman's Society (a society of wealthy Mahimal businessmen) in Kanishail. With the money handed by Mahimals, several acres of land suitable for the construction of madrasa houses, including the present government Alia Madrasa ground, located southeast of the Dargah, were purchased and the necessary construction work was also completed. Abdul Majid was questioned by some people on why he dared to approach the Mahimal community (which is generally seen as a neglected lower-class Muslim social group) for aid. He responded by saying that he did to show that this community can do big things and that they should not be neglected.
286_5	Female education was not very prevalent among the Muslims of Bengal and Assam in the past. A decade after the establishment of Sakhawat Memorial Govt. Girls' High School by Begum Rokeya, a Mahimal known as Sheikh Sikandar Ali (1891-1964) of Sheikhghat established the Muinunnisa Girls High School, named after his mother. Ali was never educated in his life though he self-taught himself and realised the value of education, and the need for the development of the uneducated Mahimal community. Initially a girls primary school, when the school was converted to a high school, the upper-class attempted to wipe away Ali and his mother's name but were unsuccessful due to protest. Ali was also the second largest benefactor of the Central Muslim Literary Society after Sareqaum Abu Zafar Abdullah. The designated monthly meeting spot for the Society was situated at Sikandar Ali's store, Anwara Woodworks, in Sheikhghat. Ali also published a weekly from 1940. Initially a mouthpiece for the Muslim
286_6	Fishermans Society of Assam Province, the magazine gained popularity among non-Muslims and Calcuttans, and was later published under the Fishermans Society of Bengal and Assam. It continued to be published until 1947 due to financial issues. Following Ali's death, poet Aminur Rashid Chowdhury wrote a lengthy editorial tribute in the Weekly Jugabheri. Ali's name is also mentioned in the Sylheter Eksho Ekjon (101 People of Sylhet) book by Captain Fazlur Rahman, author of the famous Sylheter Mati o Manush (Sylhet's land and people) history book.
286_7	Post-partition Following the Partition of India in 1947, the Mahimal communities of Bangladesh (formerly part of Pakistan) and India have developed mostly independently of each other.
286_8	Bangladesh The Mahimal community later assisted in the establishment of more madrasas in the Sylhet region such as Muhammad Ali Raipuri's Lamargaon Madrasa in Zakiganj, Bairagir Bazar Madrasa in Panchgaon and the Jamia Rahmania Taidul Islam Madrasa in Fatehpur. The latter, which hosts a science laboratory, is one of the most advanced and successful Madrasas in Bangladesh in terms of recent test results. Marhum Haji Muhammad Khurshid Ali of Bhatali, and his son, Haji Nurul Islam, greatly contributed to the establishment of the Kazir Bazar Qawmi Madrasa in Sylhet town. Another major educationist of Mahimal extraction was Haji Abdus Sattar of Haydarpur who benefacted the Bandar Bazar Jame Mosque as well as almost all major madrasas in Sylhet. Moinuddin bin Haji Bashiruddin of Kolapara Bahr successfully established a private university in the city.
286_9	The Mahimal community within Bangladesh have developed in numerous fields since independence. Notable Mahimals in the education field include: Marhum Abdul Muqit, a long-serving headmaster of the Raja GC High School Akram Ali of Sheikhpara, educated in Sylhet High Madrasa, retired Vice Principal of Madan Mohan College Afaz Uddin, professor in Sunamganj The community has also gave horizon to government secretaries such as Akmal Husayn bin Danai Haji Saheb of Dighli, Govindaganj and Zamir Uddin of Ita. High School Headmaster Ali Farid's son was former secretary of the Ministry of Home Affairs.
286_10	In India
286_11	During the early 1960s, in an attempt to emancipate this downtrodden community from the curse of socio-economic backwardness some great leaders of this community like Morhum Maulana Mumtaz Uddin, Morhum Maulana Shahid Ahmed (popularly known as Raipuri Sahib), Mr. Sarkum Ali (Master of Krishnapur, Hailakandi), Morhum Maulana Shamsul Islam, Morhum Foyez Uddin (Master Saheb of Tinghori-Bihara), Morhum Haji Sayeed Ali of Srikona (Cachar), and few others, formed an organisation called Nikhil Cachar Muslim Fishermen Federation, with an area of operation of the old Cachar district (now split into Cachar and Hailakandi). This organisation led the society to give a socio-political identity and was successful to obtain the Other Backward Classes status for the Mahimals. Since the leadership of this organisation rolled through the elderly leaders only, a few educated youths of this community, in the mid-1980s, moved to form a youth wing which was later recognised under this organisation. Large
286_12	groups of Mahimals led by the likes of Najmul Hasan, Maharam Ali (Hailakandi), Fakhar Uddin Ahmed and Abdul Noor Ahmed (Cachar) travelled across the Barak Valley, organising meetings and initiating a wave of self-identity among Mahimal youths.
286_13	Mr. Anwarul Hoque was the one and only member of Assam Legislative Assembly. (para 2–4 added by Fakhar Uddin Ahmed). The Mahimal have set up a statewide community association, the Maimal Federation, which deals with issues of community welfare. They are Sunni Muslims, and have customs similar to other Muslims of Assam in India. On the other hand, some young energetic educated boy from Maimal community have made an organization in 2012 for the allround development of said community named "Maimal Association for Humanitarian Initiative" (MAHI). Its leaders are Professor Moulana Abdul Hamid, Mohammed Abdul Waris, Ozi Uddin Ahmed, Jubayer Ahmed and others. References Social groups of Assam Muslim communities of India Fishing communities Ethnic groups in Bangladesh Fishing communities in India
287_0	The Golden State was a named passenger train between Chicago and Los Angeles from 1902–1968 on the Chicago, Rock Island and Pacific Railroad (“Rock Island”) and the Southern Pacific Company (SP) and predecessors. It was named for California, the “Golden State”. The Golden State route was relatively low-altitude, crossing the Continental Divide at about near Lordsburg, New Mexico, although the highest elevation en route was over near Corona, New Mexico. Other transcontinental routes reached elevations of more than in the Santa Fe railway near Flagstaff, Arizona, and Union Pacific near Sherman, Wyoming. At 2340 miles it was one of the longest continuous passenger railroad routes in the United States, to be exceeded by the SP's Imperial and by Amtrak's pre-2005 Sunset Limited. In most of the Arizona section of the route it passes through area acquired by the Gadsden Purchase. History
287_1	The train was inaugurated on November 2, 1902, as the Golden State Limited between Chicago, Kansas City, El Paso, southern Arizona and Los Angeles. At it had the longest route in the United States and second only to the Canadian Pacific Railway's Imperial Limited in North America. Until 1910 the Golden State Limited was seasonal, generally running December to April or May; the rest of the year, the same schedules were known as the California Limited westbound and Chicago-St. Louis Limited eastbound. The Golden State Limited was for Pullman passengers only, while the California Limited also carried tourist (economy) sleeping cars and coaches. The Golden State Limited (or California Limited in the off season) carried numbers 43 and 44 until mid-1907 when it became numbers 3 and 4. After January 1910 the Golden State Limited ran year-round until it ended in 1968. Limited was dropped from the name on May 18, 1947, and the train became the Golden State.
287_2	In summer 1926, the train left Chicago at 8:30 PM CST and arrived Los Angeles 68 hr 15 min later. During the 1920s and 1930s when Florida became a popular winter destination, the Rock Island and Southern Pacific positioned the Golden State as an escape from the cold eastern and Midwestern winters, with some success. For years the primary competition was Santa Fe's California Limited which did almost twice the business. When the Santa Fe Chief started in November 1926, the Golden State started running on the same 63-hour schedule with the same $10 extra fare (until 1929).

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