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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 Chevalley–Warning theorem ： ウィキペディア英語版 Chevalley–Warning theorem In algebra, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields . == Statement of the theorems == Let $\backslash mathbb$ be a finite field and $\backslash \_^r\backslash subseteq\backslash mathbb()$ be a set of polynomials such that the number of variables satisfies :$n>\backslash sum\_^r\; d\_j$ where $d\_j$ is the total degree of $f\_j$. The theorems are statements about the solutions of the following system of polynomial equations :$f\_j(x\_1,\backslash dots,x\_n)=0\backslash quad\backslash text\backslash ,\; j=1,\backslash ldots,\; r.$ * ''Chevalley–Warning theorem'' states that the number of common solutions $(a\_1,\backslash dots,a\_n)\; \backslash in\; \backslash mathbb^n$ is divisible by the characteristic $p$ of $\backslash mathbb$. Or in other words, the cardinality of the vanishing set of $\backslash \_^r$ is $0$ modulo $p$. * ''Chevalley's theorem'' states that if the system has the trivial solution $(0,\backslash dots,0)\; \backslash in\; \backslash mathbb^n$, i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution $(a\_1,\backslash dots,a\_n)\; \backslash in\; \backslash mathbb^n\; \backslash backslash\; \backslash$. Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since $p$ is at least 2. Both theorems are best possible in the sense that, given any $n$, the list $f\_j\; =\; x\_j,\; j=1,\backslash dots,n$ has total degree $n$ and only the trivial solution. Alternatively, using just one polynomial, we can take ''f''1 to be the degree ''n'' polynomial given by the norm of ''x''1''a''1 + ... + ''x''''n''''a''''n'' where the elements ''a'' form a basis of the finite field of order ''p''''n''. 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「Chevalley–Warning 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.