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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 Algebraic integer ： ウィキペディア英語版 Algebraic integer In number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers). The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. The ring A is the integral closure of regular integers in complex numbers. The ring of integers of a number field ''K'', denoted by ''OK'', is the intersection of ''K'' and A: it can also be characterised as the maximal order of the field ''K''. Each algebraic integer belongs to the ring of integers of some number field. A number ''x'' is an algebraic integer if and only if the ring () is finitely generated as an abelian group, which is to say, as a . ==Definitions== The following are equivalent definitions of an algebraic integer. Let ''K'' be a number field (i.e., a finite extension of $\backslash mathbb\; Q$, the set of rational numbers), in other words, $K\; =\; \backslash mathbb(\backslash theta)$ for some algebraic number $\backslash theta\; \backslash in\; \backslash mathbb$ by the primitive element theorem. * $\backslash alpha\; \backslash in\; K$ is an algebraic integer if there exists a monic polynomial $f(x)\; \backslash in\; \backslash mathbb()$ such that $f(\backslash alpha)\; =\; 0$. * $\backslash alpha\; \backslash in\; K$ is an algebraic integer if the minimal monic polynomial of $\backslash alpha$ over $\backslash mathbb\; Q$ is in $\backslash mathbb()$. * $\backslash alpha\; \backslash in\; K$ is an algebraic integer if $\backslash mathbb()$ is a finitely generated $\backslash mathbb\; Z$-module. * $\backslash alpha\; \backslash in\; K$ is an algebraic integer if there exists a finitely generated $\backslash mathbb$-submodule $M\; \backslash subset\; \backslash mathbb$ such that $\backslash alpha\; M\; \backslash subseteq\; M$. Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension $K\; /\; \backslash mathbb$. 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「Algebraic integer」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.