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In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite level" modifications of the Galois groups: if ''E''/''F'' is a finite extension, then the relative Weil group of ''E''/''F'' is ''WE''/''F'' = ''WF''/ (where the superscript ''c'' denotes the commutator subgroup). For more details about Weil groups see or or . ==Weil group of a class formation== The Weil group of a class formation with fundamental classes ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''WE''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''WE''/''F'' → Gal(''E''/''F'') → 1 corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class ''u''''E''/''F'' in ''H''2(Gal(''E''/''F''), ''A''''F''). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers ''G''/''F'', for ''F'' an open subgroup of ''G''. The reciprocity map of the class formation (''G'', ''A'') induces an isomorphism from ''AG'' to the abelianization of the Weil group. 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「Weil group」の詳細全文を読む スポンサード リンク
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