翻訳と辞書
Words near each other
・ Norman (song)
・ Norman A. Beck
・ Norman A. Erbe
・ Norman A. Fox
・ Norman A. Kent
・ Norman A. Mordue
・ Norman A. Ough
・ Norman A. Phillips
・ Norman A. Solomon
・ Norman Abbott
・ Norman Abeles
・ Norman Abrams
・ Normal degree
・ Normal distribution
・ Normal ECG
Normal extension
・ Normal family
・ Normal Field
・ Normal Field (Arizona)
・ Normal flex acres
・ Normal force
・ Normal form
・ Normal form (abstract rewriting)
・ Normal form (bifurcation theory)
・ Normal form (natural deduction)
・ Normal form for free groups and free product of groups
・ Normal function
・ Normal good
・ Normal grey cockatiel
・ Normal Happiness


Dictionary Lists
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, \mathbb(\sqrt) is a normal extension of \mathbb, since it is a splitting field of ''x''2 − 2. On the other hand, \mathbb(\sqrt()) is not a normal extension of \mathbb since the irreducible polynomial ''x''3 − 2 has one root in it (namely, \sqrt()), but not all of them (it does not have the non-real cubic roots of 2).
The fact that \mathbb(\sqrt()) is not a normal extension of \mathbb can also be seen using the first of the three properties above. The field \mathbb of algebraic numbers is an algebraic closure of \mathbb containing \mathbb(\sqrt()). On the other hand,
:\mathbb(\sqrt())=\\in\mathbb\,|\,a,b,c\in\mathbb\}
and, if ω is a primitive cubic root of unity, then the map
:\begin\sigma:&\mathbb(\sqrt())&\longrightarrow&\mathbb\\&a+b\sqrt()+c\sqrt()&\mapsto&a+b\omega\sqrt()+c\omega^2\sqrt()\end
is an embedding of \mathbb(\sqrt()) in \mathbb whose restriction to \mathbb is the identity. However, σ is not an automorphism of \mathbb(\sqrt()).
For any prime ''p'', the extension \mathbb(\sqrt(), \zeta_p) is normal of degree ''p''(''p'' − 1). It is a splitting field of ''xp'' − 2. Here \zeta_p denotes any ''p''th primitive root of unity. The field \mathbb(\sqrt(), \zeta_3) is the normal closure (see below) of \mathbb(\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.