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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 Integral extension ： ウィキペディア英語版 Integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and $a\_j\; \backslash in\; A$ such that :$b^n\; +\; a\_\; b^\; +\; \backslash cdots\; +\; a\_1\; b\; +\; a\_0\; =\; 0.$ That is to say, ''b'' is a root of a monic polynomial over ''A''.〔The above equation is sometimes called an integral equation and ''b'' is said to be integrally dependent on ''A'' (as opposed to algebraic dependent.)〕 If every element of ''B'' is integral over ''A'', then it is said that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The special case of an integral element of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g., $\backslash sqrt$). The algebraic integers in a finite extension field ''k'' of the rationals Q form a subring of ''k'', called the ring of integers of ''k'', a central object of study in algebraic number theory. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. In this article, the term ''ring'' will be understood to mean ''commutative ring'' with a multiplicative identity. ==Examples== *Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q. *Gaussian integers, complex numbers of the form $a\; +\; b\; \backslash sqrt,\; a,\; b\; \backslash in\; \backslash mathbf$, are integral over Z. $\backslash mathbf()$ is then the integral closure of Z in $\backslash mathbf(\backslash sqrt)$. *The integral closure of Z in $\backslash mathbf(\backslash sqrt)$ consists of elements of form $(a\; +\; b\; \backslash sqrt)/2$, where ''a'' and ''b'' are integers and $a^2-5b^2$ is multiple of 4; this example and the previous one are examples of quadratic integers. *Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z(). *The integral closure of Z in the field of complex numbers C is called the ''ring of algebraic integers''. *If $\backslash overline$ is an algebraic closure of a field ''k'', then $\backslash overline(\backslash dots,\; x\_n)$ is integral over $k(\backslash dots,\; x\_n).$ *Let a finite group ''G'' act on a ring ''A''. Then ''A'' is integral over ''AG'' the set of elements fixed by ''G''. see ring of invariants. *The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z. *Let ''R'' be a ring and ''u'' a unit in a ring containing ''R''. Then〔Kaplansky, 1.2. Exercise 4.〕 #''u''−1 is integral over ''R'' if and only if ''u''−1 ∈ ''R''(). #$R()\; \backslash cap\; R()$ is integral over ''R''. *The integral closure of C''x'' in a finite extension of C((''x'')) is of the form $\backslash mathbfx^$ (cf. Puiseux series) *The integral closure of the homogeneous coordinate ring of a normal projective variety ''X'' is the ring of sections ::$\backslash bigoplus\backslash nolimits\_\; \backslash operatorname^0(X,\; \backslash mathcal\_X(n)).$ 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「Integral element」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.