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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 Abel's irreducibility theorem ： ウィキペディア英語版 Abel's irreducibility theorem In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,〔.〕 asserts that if ''ƒ''(''x'') is a polynomial over a field ''F'' that shares a root with a polynomial ''g''(''x'') that is irreducible over ''F'', then every root of ''g''(''x'') is a root of ''ƒ''(''x''). Equivalently, if ''ƒ''(''x'') shares at least one root with ''g''(''x'') then ''ƒ'' is divisible evenly by ''g''(''x''), meaning that ''ƒ''(''x'') can be factored as ''g''(''x'')''h''(''x'') with ''h''(''x'') also having coefficients in ''F''.〔.〕〔This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of . Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. .〕 Corollaries of the theorem include:〔 * If ''ƒ''(''x'') is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, ''x''2 − 2 is irreducible over the rational numbers and has $\backslash sqrt$ as a root; hence there is no linear or constant polynomial over the rationals having $\backslash sqrt$ as a root. Furthermore, there is no same-degree polynomial that shares any roots with ''ƒ''(''x''), other than constant multiples of ''ƒ''(''x''). * If ''ƒ''(''x'') ≠ ''g''(''x'') are two different irreducible monic polynomials, then they share no roots. == References == 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「Abel'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.