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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 Hensel's lemma ： ウィキペディア英語版 Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power of , which can be found by iteratively "lifting" the solution modulo successive powers of . More generally it is used as a generic name for analogues for complete commutative rings (including ''p''-adic fields in particular) of the Newton method for solving equations. Since ''p''-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial. == Statement == Let $f(x)$ be a polynomial with integer (or ''p''-adic integer) coefficients, and let ''m'',''k'' be positive integers such that ''m'' ≤ ''k''. If ''r'' is an integer such that :$f(r)\; \backslash equiv\; 0\; \backslash pmod$ and $f\text{'}(r)\; \backslash not\backslash equiv\; 0\; \backslash pmod$ then there exists an integer ''s'' such that :$f(s)\; \backslash equiv\; 0\; \backslash pmod\}.$ Furthermore, this ''s'' is unique modulo ''p''''k''+m, and can be computed explicitly as :$s\; =\; r\; +\; tp^k$ where $t\; =\; -\; \backslash frac\; \backslash cdot\; (f\text{'}(r)^).$ In this formula for ''t'', the division by ''p''''k'' denotes ordinary integer division (where the remainder will be 0), while negation, multiplication, and multiplicative inversion $f\text{'}(r)^$ are performed in $\backslash mathbb/p^m\backslash mathbb$. As an aside, if $f\text{'}(r)\; \backslash equiv\; 0\; \backslash pmod$, then 0, 1, or several ''s'' may exist (see Hensel Lifting below). 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「Hensel's lemma」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.