翻訳と辞書 Words near each other ・ Hasse Andersson ・ Hasse Aro ・ Hasse Borg ・ Hasse Börjes ・ Hasse derivative ・ Hasse diagram ・ Hasse Ekman ・ Hasse Fröberg ・ Hasse Funck ・ Hasse invariant ・ Hasse invariant of a quadratic form ・ Hasse invariant of an algebra ・ Hasse Jeppson ・ Hasse Lake (Alberta) ・ Hasse Mattisson ・ Hasse norm theorem ・ Hasse Pavia Lind ・ Hasse Persson ・ Hasse principle ・ Hasse Sjöö ・ Hasse Thomsén ・ Hasse Walli ・ Hasse Zetterström ・ Hasse's theorem ・ Hasse's theorem on elliptic curves ・ Hassea ・ Hassegau ・ Hasseh ・ Hassei Takano ・ Hassein Ismail Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hasse norm theorem ： ウィキペディア英語版 Hasse norm theorem In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element ''k'' of K such that there is an element ''l'' of L with $\backslash mathbf\_(l)\; =\; k$; in other words ''k'' is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then ''k'' is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension $(\backslash sqrt,\backslash sqrt)/$ but is not a global norm. Serre and Tate showed that another counterexample is given by the field $(\backslash sqrt,\backslash sqrt)/$ where every rational square is a local norm everywhere but $5^2$ is not a global norm. This is an example of a theorem stating a local-global principle. The full theorem is due to . The special case when the degree ''n'' of the extension is 2 was proved by , and the special case when ''n'' is prime was proved by . The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(''L''/''K'') is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(''L''/''K'') is isomorphic to the Tate cohomology group H0(''L''/''K'') which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere. ==See also== *Grunwald–Wang theorem, about when an element that is a power everywhere locally is a power. 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「Hasse norm 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.