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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 Power residue symbol ： ウィキペディア英語版 Power residue symbol In algebraic number theory the ''n''-th power residue symbol (for an integer ''n'' > 2) is a generalization of the (quadratic) Legendre symbol to ''n''-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher〔Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.〕 reciprocity laws.〔All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2〕 ==Background and notation== Let ''k'' be an algebraic number field with ring of integers   $\backslash mathcal\_k$  that contains a primitive ''n''-th root of unity   $\backslash zeta\_n\backslash in\backslash mathcal\_k.$ Let   $\backslash mathfrak\; \backslash subset\; \backslash mathcal\_k$   be a prime ideal and assume that ''n'' and $\backslash mathfrak$ are coprime (i.e. $n\; \backslash not\; \backslash in\; \backslash mathfrak$.) The norm of  $\backslash mathfrak$  is defined as the cardinality of the residue class ring   $\backslash mathcal\_k\; /\; \backslash mathfrak\backslash ;:\backslash ;\backslash ;\backslash ;\; \backslash mathrm\; \backslash mathfrak\; =\; |\backslash mathcal\_k\; /\; \backslash mathfrak|.$   (since $\backslash mathfrak$ is prime this is a finite field) There is an analogue of Fermat's theorem in   $\backslash mathcal\_k:$  If   $\backslash alpha\; \backslash in\; \backslash mathcal\_k,\backslash ;\backslash ;\backslash ;\; \backslash alpha\backslash not\backslash in\; \backslash mathfrak,$   then :$\backslash alpha^\; -1\}\backslash equiv\; 1\; \backslash pmod.$ And finally,   $\backslash mathrm\; \backslash mathfrak\; \backslash equiv\; 1\; \backslash pmod.$   These facts imply that :$\backslash alpha^\; -1\}\}\backslash equiv\; \backslash zeta\_n^s\backslash pmod$   is well-defined and congruent to a unique ''n''-th root of unity ζ''n''''s''. 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「Power residue symbol」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.