翻訳と辞書 Words near each other ・ Hilbert transform ・ Hilbert van der Duim ・ Hilbert Van Dijk ・ Hilbert Wildlife Management Area ・ Hilbert's arithmetic of ends ・ Hilbert's axioms ・ Hilbert's basis theorem ・ Hilbert's eighteenth problem ・ Hilbert's eighth problem ・ Hilbert's eleventh problem ・ Hilbert's fifteenth problem ・ Hilbert's fifth problem ・ Hilbert's fourteenth problem ・ Hilbert's fourth problem ・ Hilbert's inequality ・ Hilbert's irreducibility theorem ・ Hilbert's lemma ・ Hilbert's nineteenth problem ・ Hilbert's ninth problem ・ Hilbert's Nullstellensatz ・ Hilbert's paradox of the Grand Hotel ・ Hilbert's problems ・ Hilbert's program ・ Hilbert's second problem ・ Hilbert's seventeenth problem ・ Hilbert's seventh problem ・ Hilbert's sixteenth problem ・ Hilbert's sixth problem ・ Hilbert's syzygy theorem ・ Hilbert's tenth problem Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert's irreducibility theorem ： ウィキペディア英語版 Hilbert's irreducibility theorem In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory. == Formulation of the theorem == Hilbert's irreducibility theorem. Let : $f\_1(X\_1,\backslash ldots,\; X\_r,\; Y\_1,\backslash ldots,\; Y\_s),\; \backslash ldots,\; f\_n(X\_1,\backslash ldots,\; X\_r,\; Y\_1,\backslash ldots,\; Y\_s)\; \backslash ,$ be irreducible polynomials in the ring :$\backslash mathbb(X\_r,\; Y\_1,\backslash ldots,\; Y\_s).\; \backslash ,$ Then there exists an ''r''-tuple of rational numbers (''a''1,...,''a''''r'') such that : $f\_1(a\_1,\backslash ldots,\; a\_r,\; Y\_1,\backslash ldots,\; Y\_s),\; \backslash ldots,\; f\_n(a\_1,\backslash ldots,\; a\_r,\; Y\_1,\backslash ldots,\; Y\_s)\; \backslash ,$ are irreducible in the ring :$\backslash mathbb(Y\_s).\; \backslash ,$ Remarks. * It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specialization, called Hilbert set, is large in many senses. For example, this set is Zariski dense in $\backslash mathbb\; Q^r$. * There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''1,...,''a''''r'') to be integers. * There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fields are Hilbertian.〔Lang (1997) p.41〕 * The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take $n=r=s=1$ in the definition. A recent result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of $n=r=s=1$ and $f=f\_1$ absolutely irreducible, that is, irreducible in the ring ''K''alg(), where ''K''alg is the algebraic closure of ''K''. 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「Hilbert's irreducibility 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.