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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. ==Definitions== 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''''E'' is defined to be the elements of ''A'' fixed by ''E''. We write :''H''''n''(''E''/''F'') for the Tate cohomology group ''H''''n''(''E''/''F'', ''A''''F'') 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''1(''E''/''F'') is trivial, and :''H''2(''E''/''F'') is cyclic of order |''E''/''F''|. In practice, these cyclic groups come provided with canonical generators ''u''''E''/''F'' ∈ ''H''2(''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''1(''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「class formation」の詳細全文を読む スポンサード リンク
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