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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 Primitive element theorem ： ウィキペディア英語版 Primitive element theorem In field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions. It says that a finite extension is simple if and only if there are only finitely many intermediate fields. In particular, finite separable extensions are simple. == Terminology == Let $E\backslash supseteq\; F$ be a finite field extension. An element $\backslash alpha\backslash in\; E$ is said to be a ''primitive element'' for $E\backslash supseteq\; F$ when :$E=F(\backslash alpha).$ In this situation, the extension $E\backslash supseteq\; F$ is referred to as a ''simple extension''. Then every element ''x'' of ''E'' can be written in the form :$x=f\_^+\backslash cdots+f\_1+f\_0,$ where $f\_i\backslash in\; F$ for all ''i'', and $\backslash alpha\backslash in\; E$ is fixed. That is, if $E\backslash supseteq\; F$ is separable of degree ''n'', there exists $\backslash alpha\backslash in\; E$ such that the set :$\backslash \backslash \}$ is a basis for ''E'' as a vector space over ''F''. For instance, the extensions $\backslash mathbb(\backslash sqrt)\backslash supseteq\; \backslash mathbb$ and $\backslash mathbb(x)\backslash supseteq\; \backslash mathbb$ are simple extensions with primitive elements $\backslash sqrt$ and ''x'', respectively ($\backslash mathbb(x)$ denotes the field of rational functions in the indeterminate ''x'' over $\backslash mathbb$). 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「Primitive element 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.