翻訳と辞書 Words near each other ・ Stick to Your Guns (film) ・ Stick to Your Vision ・ Stick Together ・ Stick with Me, Kid ・ Stick-built home ・ Stick-fighting ・ Stick-in-the-jug ・ Stick-slip phenomenon ・ Stick-Up ・ Stick-Up! ・ Stickam ・ Stickball ・ Stickball (album) ・ Stickeen ・ Stickeen Territories ・ Stickelberger's theorem ・ Stickelgraben ・ Stickens ・ Sticker ・ Sticker (disambiguation) ・ Sticker (Internet) ・ Sticker album ・ Sticker art ・ Sticker grass ・ Sticker Happy ・ Sticker hårt ・ Sticker shock ・ Sticker, Cornwall ・ Stickers as a Service ・ Stickets 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 Stickelberger's theorem ： ウィキペディア英語版 Stickelberger's theorem In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer () while the general result is due to Ludwig Stickelberger (). ==The Stickelberger element and the Stickelberger ideal== Let ''Km'' denote the ''m''th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the ''m''th roots of unity to Q (where ''m'' ≥ 2 is an integer). It is a Galois extension of Q with Galois group ''Gm'' isomorphic to the multiplicative group of integers modulo ''m'' (Z/''m''Z)×. The Stickelberger element (of level ''m'' or of ''Km'') is an element in the group ring Q() and the Stickelberger ideal (of level ''m'' or of ''Km'') is an ideal in the group ring Z(). They are defined as follows. Let ζ''m'' denote a primitive ''m''th root of unity. The isomorphism from (Z/''m''Z)× to ''Gm'' is given by sending ''a'' to σ''a'' defined by the relation :σ''a''(ζ''m'') = . The Stickelberger element of level ''m'' is defined as :$\backslash theta(K\_m)=\backslash frac\backslash underseta\backslash cdot\backslash sigma\_a^\backslash in\backslash mathbf().$ The Stickelberger ideal of level ''m'', denoted ''I''(''Km''), is the set of integral multiples of θ(''Km'') which have integral coefficients, i.e. :$I(K\_m)=\backslash theta(K\_m)\backslash mathbf()\backslash cap\backslash mathbf().$ More generally, if ''F'' be any abelian number field whose Galois group over Q is denoted ''GF'', then the Stickelberger element of ''F'' and the Stickelberger ideal of ''F'' can be defined. By the Kronecker–Weber theorem there is an integer ''m'' such that ''F'' is contained in ''Km''. Fix the least such ''m'' (this is the (finite part of the) conductor of ''F'' over Q). There is a natural group homomorphism ''Gm'' → ''GF'' given by restriction, i.e. if σ ∈ ''Gm'', its image in ''GF'' is its restriction to ''F'' denoted res''m''σ. The Stickelberger element of ''F'' is then defined as :$\backslash theta(F)=\backslash frac\backslash underseta\backslash cdot\backslash mathrm\_m\backslash sigma\_a^\backslash in\backslash mathbf().$ The Stickelberger ideal of ''F'', denoted ''I''(''F''), is defined as in the case of ''Km'', i.e. :$I(F)=\backslash theta(F)\backslash mathbf()\backslash cap\backslash mathbf().$ In the special case where ''F'' = ''Km'', the Stickelberger ideal ''I''(''Km'') is generated by (''a'' − σ''a'')θ(''Km'') as ''a'' varies over Z/''m''Z. This not true for general ''F''.〔, Lemma 6.9 and the comments following it〕 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「Stickelberger's 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.