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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 Hasse–Davenport relation ： ウィキペディア英語版 Hasse–Davenport relation The Hasse–Davenport relations, introduced by , are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory relating Gauss sums over different fields. used it to calculate the zeta function of a Fermat hypersurface over a finite field, which motivated the Weil conjectures. Gauss sums are analogues of the gamma function over finite fields, and the Hasse–Davenport product relation is the analogue of Gauss's multiplication formula :$$ \Gamma(z) \; \Gamma\left(z + \frac\right) \; \Gamma\left(z + \frac\right) \cdots \Gamma\left(z + \frac\right) = (2 \pi)^} \; k^ \; \Gamma(kz). \,\! In fact the Hasse–Davenport product relation follows from the analogous multiplication formula for ''p''-adic gamma functions together with the Gross–Koblitz formula of . == Hasse–Davenport lifting relation == Let ''F'' be a finite field with ''q'' elements, and ''F''s be the field such that () = ''s'', that is, ''s'' is the dimension of the vector space ''F''s over ''F''. Let $\backslash alpha$ be an element of $F\_s$. Let $\backslash chi$ be a multiplicative character from ''F'' to the complex numbers. Let $N\_(\backslash alpha)$ be the norm from $F\_s$ to $F$ defined by :$N\_(\backslash alpha):=\backslash alpha\backslash cdot\backslash alpha^q\backslash cdots\backslash alpha^(\backslash alpha))$ Let ψ be some nontrivial additive character of ''F'', and let $\backslash psi\text{'}$ be the additive character on $F\_s$ which is the composition of $\backslash psi$ with the trace from ''F''s to ''F'', that is :$\backslash psi\text{'}(\backslash alpha):=\backslash psi(Tr\_(\backslash alpha))$ Let :$\backslash tau(\backslash chi,\backslash psi)=\backslash sum\_\backslash chi(x)\backslash psi(x)$ be the Gauss sum over ''F'', and let $\backslash tau(\backslash chi\text{'},\backslash psi\text{'})$ be the Gauss sum over $F\_s$. Then the Hasse–Davenport lifting relation states that :$(-1)^s\backslash cdot\; \backslash tau(\backslash chi,\backslash psi)^s=-\backslash tau(\backslash chi\text{'},\backslash psi\text{'}).$ 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–Davenport relation」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.