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Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-abelian Class Field Theory
In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field ''K'', to the general Galois extension ''L''/''K''. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense. History A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the idele class group. The theorems of the cohomological approach are independent of whether or not the Galois group ''G'' of ''L''/''K'' is abelian. This theory has never been regarded as the sought-after ''non-abelian'' theory. The first reason that can be cited for that is that it did not provide fresh information on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Group
Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Brauer (1929–2021), Austrian painter, poet, and actor, father of Timna Brauer * August Brauer (1863-1917), German zoologist * Friedrich Moritz Brauer (1832–1904), Austrian entomologist and museum director * Georg Brauer (1908–2001), German chemist * Ingrid Arndt-Brauer (born 1961), German politician; member of the Bundestag * Jono Brauer (born 1981), Australian Olympic skier * Max Brauer (1887–1973), German politician; First Mayor of Hamburg * Michael Brauer (contemporary), American audio engineer * Rich Brauer (born 1954), American politician from Illinois; state legislator since 2003 * Richard Brauer (1901–1977), German-American mathematician * Richard H. W. Brauer (contemporary), American art museum director; eponym of the Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmy Noether
Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile aphysikerinnen.de/noetherlebenslauf.html). Sometimes ''Emmy'' is mistakenly reported as a short form for ''Amalie'', or misreported as "Emily". e.g. (, ; ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's First and Second Theorem, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Map
Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may also refer to: Mathematics, science and technology Computing * IBM POWER (software), an IBM operating system enhancement package * IBM POWER architecture, a RISC instruction set architecture * Power ISA, a RISC instruction set architecture derived from PowerPC * IBM Power microprocessors, made by IBM, which implement those RISC architectures * Power.org, a predecessor to the OpenPOWER Foundation * SGI POWER Challenge, a line of SGI supercomputers Mathematics * Exponentiation, "''x'' to the power of ''y''" * Power function * Power of a point * Statistical power Physics * Magnification, the factor by which an optical system enlarges an image * Optical power, the degree to which a lens converges or diverges light Social sciences and politi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer ''n'' – and therefore belong to abstract algebra. The theory of cyclic extensions of the field ''K'' when the characteristic of ''K'' does divide ''n'' is called Artin–Schreier theory. Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of fields with cyclic Galois group ''G'' = Gal(''L''/''K'') generated by an element \sigma, and if a is an element of ''L'' of relative norm 1, that isN(a):=a\, \sigma(a)\, \sigma^2(a)\cdots \sigma^(a)=1,then there exists b in ''L'' such thata=b/\sigma(b).The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht , although it is originally due to . Often a more general theorem due to is given the name, stating that if ''L''/''K'' is a finite Galois extension of fields with arbitrary Galois group ''G'' = Gal(''L''/''K''), then the first cohomology group of ''G'', with coefficients in the multiplicative group of ''L'', is trivial: :H^1(G,L^\times)=\. Examples Let L/K be the quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field ''F'', the group is , where 0 refers to the zero element of ''F'' and the binary operation • is the field multiplication, *the algebraic torus GL(1).. Examples *The multiplicative group of integers modulo ''n'' is the group under multiplication of the invertible elements of \mathbb/n\mathbb. When ''n'' is not prime, there are elements other than zero that are not invertible. * The multiplicative group of positive real numbers \mathbb^+ is an abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, \mathbb. * The multiplicative group of a field F is the set of all nonzero elements: F^\times = F -\, under the multiplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as ''Logicism''. Life Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. He first attended the Collegium Carolinum in 1848 before transferring to the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Group Of A Ring
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the ''additive group'' of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that i ...s. References {{DEFAULTSORT:Additive group Algebraic structures Group theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
