Articoli correlati a Class Field Theory
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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Class Formation, Artin Reciprocity Law, Artin L-Function, Conductor, Langlands Group, Iwasawa Theory, Class Field Theory, Hilbert Class Field, Complex Multiplication, Takagi Existence Theorem, Hilbert Symbol, Galois Cohomology, Golod-shafarevich Theorem, Tate Cohomology Group, Non-Abelian Class Field Theory, Quasi-Finite Field, Local Class Field Theory, Local Fields, Abelian Extension, Kronecker-weber Theorem, Hasse Norm Theorem, Genus Field. Excerpt: In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian . When the Galois group is a cyclic group, we have a cyclic extension . Any finite extension of a finite field is a cyclic extension. The development of class field theory has provided detailed information about abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields . There are two slightly different concepts of cyclotomic extension s: these can mean either extensions formed by adjoining roots of unity, or subextensions of such extensions. The cyclotomic fields are examples. Any cyclotomic extension (for either definition) is abelian. If a field K contains a primitive n -th root of unity and the n -th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension ). In general, however, the Galois groups of n -th roots of elements operate both on the n -th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product . The Kummer theory gives a complete description of the abelian extension case, and the Kronecker Weber theorem tells us that if K is the field of rational numbers, an exte...
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Class field theory
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Books LLC, Reference Series Mai 2020, 2020
ISBN 10: 1155340582
ISBN 13: 9781155340586
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Da: buchversandmimpf2000, Emtmannsberg, BAYE, Germania
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Taschenbuch. Condizione: Neu. Neuware -Source: Wikipedia. Pages: 24. Chapters: Abelian extension, Albert¿Brauer¿Hasse¿Noether theorem, Artin L-function, Artin reciprocity law, Class formation, Complex multiplication, Conductor (class field theory), Galois cohomology, Genus field, Golod¿Shafarevich theorem, Grunwald¿Wang theorem, Hasse norm theorem, Hilbert class field, Hilbert symbol, Iwasawa theory, Kronecker¿Weber theorem, Lafforgue's theorem, Langlands dual, Langlands¿Deligne local constant, Local class field theory, Local Fields (book), Local Langlands conjectures, Non-abelian class field theory, Quasi-finite field, Takagi existence theorem, Tate cohomology group, Weil group. Excerpt: In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory. A formation is a topological group G together with a topological G-module A on which G acts continuously. A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then A is defined to be the elements of A fixed by E. We write H(E/F)for the Tate cohomology group H(E/F, A) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.) In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. A class formation is a formation such that for every normal layer E/F H(E/F) is trivial, andH(E/F) is cyclic of order |E/F|.In practice, these cyclic groups come provided with canonical generators uE/F ¿ H(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation. A formation that satisfies just the condition H(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert's theorem 90. The most important examples of class formations (arranged roughly in order of difficulty) are as follows: It is easy to verify the class formation property for the finite field case and the archimedean local field case, butBooks on Demand GmbH, Überseering 33, 22297 Hamburg 24 pp. Englisch. Codice articolo 9781155340586
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