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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 Purely inseparable extension ： ウィキペディア英語版 Purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x''''q'' = ''a'', with ''q'' a power of ''p'' and ''a'' in ''k''. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. ==Purely inseparable extensions== An algebraic extension $E\backslash supseteq\; F$ is a ''purely inseparable extension'' if and only if for every $\backslash alpha\backslash in\; E\backslash setminus\; F$, the minimal polynomial of $\backslash alpha$ over ''F'' is ''not'' a separable polynomial.〔Isaacs, p. 298〕 If ''F'' is any field, the trivial extension $F\backslash supseteq\; F$ is purely inseparable; for the field ''F'' to possess a ''non-trivial'' purely inseparable extension, it must be imperfect as outlined in the above section. Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If $E\backslash supseteq\; F$ is an algebraic extension with (non-zero) prime characteristic ''p'', then the following are equivalent:〔Isaacs, Theorem 19.10, p. 298〕 1. ''E'' is purely inseparable over ''F.'' 2. For each element $\backslash alpha\backslash in\; E$, there exists $n\backslash geq\; 0$ such that $\backslash alpha^\backslash in\; F$. 3. Each element of ''E'' has minimal polynomial over ''F'' of the form $X^-a$ for some integer $n\backslash geq\; 0$ and some element $a\backslash in\; F$. It follows from the above equivalent characterizations that if $E=F()$ (for ''F'' a field of prime characteristic) such that $\backslash alpha^\backslash in\; F$ for some integer $n\backslash geq\; 0$, then ''E'' is purely inseparable over ''F''.〔Isaacs, Corollary 19.11, p. 298〕 (To see this, note that the set of all ''x'' such that $x^\backslash in\; F$ for some $n\backslash geq\; 0$ forms a field; since this field contains both $\backslash alpha$ and ''F'', it must be ''E'', and by condition 2 above, $E\backslash supseteq\; F$ must be purely inseparable.) If ''F'' is an imperfect field of prime characteristic ''p'', choose $a\backslash in\; F$ such that ''a'' is not a ''p''th power in ''F'', and let ''f''(''X'') = ''X''p − ''a''. Then ''f'' has no root in ''F'', and so if ''E'' is a splitting field for ''f'' over ''F'', it is possible to choose $\backslash alpha$ with $f(\backslash alpha)=0$. In particular, $\backslash alpha^=a$ and by the property stated in the paragraph directly above, it follows that $F()\backslash supseteq\; F$ is a non-trivial purely inseparable extension (in fact, $E=F()$, and so $E\backslash supseteq\; F$ is automatically a purely inseparable extension).〔Isaacs, p. 299〕 Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If ''K'' is a field of characteristic ''p'', and if ''V'' is an algebraic variety over ''K'' of dimension greater than zero, the function field ''K''(''V'') is a purely inseparable extension over the subfield ''K''(''V'')''p'' of ''p''th powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by ''p'' on an elliptic curve over a finite field of characteristic ''p''. 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「Purely inseparable extension」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.