Class field theory

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, but