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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 Adele ring ： ウィキペディア英語版 Adele ring In algebraic number theory and topological algebra, the adele ring〔Also spelled: ''adèle ring'' .〕 (other names are the adelic ring, the ring of adeles) is a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field). It involves in a symmetric way all the completions of the field. The adele ring was introduced by Claude Chevalley for the purposes of simplifying and clarifying class field theory. It has also found applications outside that area. The adele ring and its relation to the number field are among the most fundamental objects in number theory. The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory. It is a central principle of Diophantine geometry to study solutions of polynomial equations in number fields by looking at their solutions in the larger complete adele ring, where it is generally easier to detect solutions, and then deciding which of them come from the number field. The word "adele" is short for "additive idele"〔Neukirch (1999) p. 357.〕 and it was invented by André Weil. The previous name was the valuation vectors. The ring of adeles was historically preceded by the ring of repartitions, a construction which avoids completions, and is today sometimes referred to as pre-adele. == Definitions == The profinite completion of the integers, $\backslash widehat\; /\; n\; \backslash mathbb$: :$\backslash widehat/n\backslash mathbb.$ By the Chinese remainder theorem it is isomorphic to the product of all the rings of ''p''-adic integers: :$\backslash widehat\; \backslash mathbb\_p.$ The ring of integral adeles AZ is the product :$\backslash mathbb\_\backslash mathbb\; =\; \backslash mathbb\; \backslash times\; \backslash widehat\_\backslash mathbb\; =\backslash mathbb\backslash otimes\_\backslash mathbb\; Z\; \backslash mathbb\_\backslash mathbb$ (topologized so that AZ is an open subring). More generally the ring of adeles A''F'' of any algebraic number field ''F'' is the tensor product :$\backslash mathbb\_F\; =F\backslash otimes\_\backslash mathbb\; Z\; \backslash mathbb\_\backslash mathbb$ (topologized as the product of $\backslash deg(F)$ copies of AQ). The ring of (rational) adeles can also be defined as the restricted product :$\backslash mathbb\_\backslash mathbb\; =\; \backslash mathbb\; \backslash times\; \_p$ of all the ''p''-adic completions Q''p'' and the real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele (''a''∞, ''a''2, ''a''3, ''a''5, …) all but a finite number of the ''a''''p'' are ''p''-adic integers.〔 The adeles of a function field over a finite field can be defined in a similar way, as the restricted product of all completions. 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「Adele ring」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.