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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Shafarevich–Weil theorem ： ウィキペディア英語版 Shafarevich–Weil theorem In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by for local fields and by for global fields. ==Statement== Suppose that ''F'' is a global field, ''K'' is a normal extension of ''F'', and ''L'' is an abelian extension of ''K''. Then the Galois group Gal(''L''/''F'') is an extension of the group Gal(''K''/''F'') by the abelian group Gal(''L''/''K''), and this extension corresponds to an element of the cohomology group H2(Gal(''K''/''F''), Gal(''L''/''K'')). On the other hand, class field theory gives a fundamental class in H2(Gal(''K''/''F''),''I''''K'') and a reciprocity law map from ''I''''K'' to Gal(''L''/''K''). The Shafarevich–Weil theorem states that the class of the extension Gal(''L''/''F'') is the image of the fundamental class under the homomorphism of cohomology groups induced by the reciprocity law map . Shafarevich stated his theorem for local fields in terms of division algebras rather than the fundamental class . In this case, with ''L'' the maximal abelian extension of ''K'', the extension Gal(''L''/''F'') corresponds under the reciprocity map to the normalizer of ''K'' in a division algebra of degree () over ''F'', and Shafarevich's theorem states that the Hasse invariant of this division algebra is 1/(). The relation to the previous version of the theorem is that division algebras correspond to elements of a second cohomology group (the Brauer group) and under this correspondence the division algebra with Hasse invariant 1/() corresponds to the fundamental class. 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「Shafarevich–Weil theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.