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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 Conductor (class field theory) ： ウィキペディア英語版 Conductor (class field theory) In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map. ==Local conductor== Let ''L''/''K'' be a finite abelian extension of non-archimedean local fields. The conductor of ''L''/''K'', denoted $\backslash mathfrak(L/K)$, is the smallest non-negative integer ''n'' such that the higher unit group :$U^=1+\backslash mathfrak^n=\backslash left\backslash \backslash ,\backslash mathfrak\_K^n)\backslash right\backslash \}$ is contained in ''NL''/''K''(''L''×), where ''NL''/''K'' is field norm map and $\backslash mathfrak\_K$ is the maximal ideal of ''K''. Equivalently, ''n'' is the smallest integer such that the local Artin map is trivial on $U\_K^$. Sometimes, the conductor is defined as $\backslash mathfrak\_K^n$ where ''n'' is as above.〔As in 〕 The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero, and it is tamely ramified if, and only if, the conductor is 1. More precisely, the conductor computes the non-triviality of higher ramification groups: if ''s'' is the largest integer for which the "lower numbering" higher ramification group ''Gs'' is non-trivial, then $\backslash mathfrak(L/K)=\backslash eta\_(s)+1$, where η''L''/''K'' is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups. The conductor of ''L''/''K'' is also related to the Artin conductors of characters of the Galois group Gal(''L''/''K''). Specifically, :$\backslash mathfrak\_K^=\backslash underset\_K^$ where χ varies over all multiplicative complex characters of Gal(''L''/''K''), $\backslash mathfrak\_\backslash chi$ is the Artin conductor of χ, and lcm is the least common multiple. 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「Conductor (class field theory)」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.