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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 Normal extension ： ウィキペディア英語版 Normal extension In abstract algebra, an algebraic field extension ''L''/''K'' is said to be normal if ''L'' is the splitting field of a family of polynomials in ''K''(). Bourbaki calls such an extension a quasi-Galois extension. == Equivalent properties and examples == The normality of ''L''/''K'' is equivalent to either of the following properties. Let ''K''''a'' be an algebraic closure of ''K'' containing ''L''. * Every embedding σ of ''L'' in ''K''''a'' that restricts to the identity on ''K'', satisfies σ(''L'') = ''L''. In other words, σ is an automorphism of ''L'' over ''K''. * Every irreducible polynomial in ''K''() that has one root in ''L'', has all of its roots in ''L'', that is, it decomposes into linear factors in ''L''(). (One says that the polynomial ''splits'' in ''L''.) If ''L'' is a finite extension of ''K'' that is separable (for example, this is automatically satisfied if ''K'' is finite or has characteristic zero) then the following property is also equivalent: * There exists an irreducible polynomial whose roots, together with the elements of ''K'', generate ''L''. (One says that ''L'' is the splitting field for the polynomial.) For example, $\backslash mathbb(\backslash sqrt)$ is a normal extension of $\backslash mathbb$, since it is a splitting field of ''x''2 − 2. On the other hand, $\backslash mathbb(\backslash sqrt())$ is not a normal extension of $\backslash mathbb$ since the irreducible polynomial ''x''3 − 2 has one root in it (namely, $\backslash sqrt()$), but not all of them (it does not have the non-real cubic roots of 2). The fact that $\backslash mathbb(\backslash sqrt())$ is not a normal extension of $\backslash mathbb$ can also be seen using the first of the three properties above. The field $\backslash mathbb$ of algebraic numbers is an algebraic closure of $\backslash mathbb$ containing $\backslash mathbb(\backslash sqrt())$. On the other hand, :$\backslash mathbb(\backslash sqrt())=\backslash \backslash in\backslash mathbb\backslash ,|\backslash ,a,b,c\backslash in\backslash mathbb\backslash \}$ and, if ω is a primitive cubic root of unity, then the map : is an embedding of $\backslash mathbb(\backslash sqrt())$ in $\backslash mathbb$ whose restriction to $\backslash mathbb$ is the identity. However, σ is not an automorphism of $\backslash mathbb(\backslash sqrt())$. For any prime ''p'', the extension $\backslash mathbb(\backslash sqrt(),\; \backslash zeta\_p)$ is normal of degree ''p''(''p'' − 1). It is a splitting field of ''xp'' − 2. Here $\backslash zeta\_p$ denotes any ''p''th primitive root of unity. The field $\backslash mathbb(\backslash sqrt(),\; \backslash zeta\_3)$ is the normal closure (see below) of $\backslash mathbb(\backslash sqrt())$. 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「Normal 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.