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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 Fundamental theorem of Galois theory ： ウィキペディア英語版 Fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. In its most basic form, the theorem asserts that given a field extension ''E''/''F'' that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields are fields ''K'' satisfying ''F'' ⊆ ''K'' ⊆ ''E''; they are also called ''subextensions'' of ''E''/''F''.) ==Explicit description of the correspondence== For finite extensions, the correspondence can be described explicitly as follows. * For any subgroup ''H'' of Gal(''E''/''F''), the corresponding fixed field, denoted ''EH'', is the set of those elements of ''E'' which are fixed by every automorphism in ''H''. * For any intermediate field ''K'' of ''E''/''F'', the corresponding subgroup is Aut(''E''/''K''), that is, the set of those automorphisms in Gal(''E''/''F'') which fix every element of ''K''. The fundamental theorem says that this correspondence is a one-to-one correspondence if (and only if) ''E''/''F'' is a Galois extension. For example, the topmost field ''E'' corresponds to the trivial subgroup of Gal(''E''/''F''), and the base field ''F'' corresponds to the whole group Gal(''E''/''F''). The notation Gal(''E''/''F'') is only used for Galois extensions. If ''E''/''F'' is Galois, then Gal(''E''/''F'') = Aut(''E''/''F''). If ''E''/''F'' is not Galois, then the "correspondence" gives only an injective (but not surjective) map from $\backslash$ to $\backslash$, and a surjective (but not injective) map in the reverse direction. In particular, if ''E''/''F'' is not Galois, then ''F'' is not the fixed field of any subgroup of Aut(''E''/''F''). 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「Fundamental theorem of Galois theory」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.