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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 Cohn's irreducibility criterion ： ウィキペディア英語版 Cohn's irreducibility criterion Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in $\backslash mathbb()$—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients. The criterion is often stated as follows: :If a prime number $p$ is expressed in base 10 as $p=a\_m10^m+a\_10^+\backslash cdots+a\_110+a\_0$ (where $0\backslash leq\; a\_i\backslash leq\; 9$) then the polynomial ::$f(x)=a\_mx^m+a\_x^+\backslash cdots+a\_1x+a\_0$ :is irreducible in $\backslash mathbb()$. The theorem can be generalized to other bases as follows: :Assume that $b\; \backslash ge\; 2$ is a natural number and $p(x)=a\_kx^k+a\_x^+\backslash cdots+a\_1x+a\_0$ is a polynomial such that $0\backslash leq\; a\_i\backslash leq\; b-1$. If $p(b)$ is a prime number then $p(x)$ is irreducible in $\backslash mathbb()$. The base-10 version of the theorem attributed to Cohn by Pólya and Szegő in one of their books〔 English translation in: 〕 while the generalization to any base, 2 or greater, is due to Brillhart, Filaseta, and Odlyzko. In 2002, Ram Murty gave a simplified proof as well as some history of the theorem in a paper that is available online.〔 (dvi file)〕 The converse of this criterion is that, if ''p'' is an irreducible polynomial with integer coefficients that have greatest common divisor 1, then there exists a base such that the coefficients of ''p'' form the representation of a prime number in that base; this is the Bunyakovsky conjecture and its truth or falsity remains an open question. ==Historical notes== *Polya and Szegő gave their own generalization but it has many side conditions (on the locations of the roots, for instance) so it lacks the elegance of Brillhart's, Filaseta's, and Odlyzko's generalization. *It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn (1894–1940), a student of Issai Schur who was awarded his PhD in Berlin in 1921.〔(Arthur Cohn's entry at the Mathematics Genealogy Project )〕 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「Cohn's irreducibility criterion」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.