|
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ''F''. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. 〔 See the article Galois group for definitions of some of these terms and some examples.〕 A result of Emil Artin allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'' with fixed field ''F'', then ''E''/''F'' is a Galois extension. ==Characterization of Galois extensions== An important theorem of Emil Artin states that for a finite extension ''E''/''F'', each of the following statements is equivalent to the statement that ''E''/''F'' is Galois: * ''E''/''F'' is a normal extension and a separable extension. * ''E'' is a splitting field of a separable polynomial with coefficients in ''F''. * |Aut(''E''/''F'')| = (), that is, the number of automorphisms equals the degree of the extension. Other equivalent statements are: * Every irreducible polynomial in ''F''() with at least one root in ''E'' splits over ''E'' and is separable. * |Aut(''E''/''F'')| ≥ (), that is, the number of automorphisms is at least the degree of the extension. * ''F'' is the fixed field of a subgroup of Aut(''E''). * ''F'' is the fixed field of Aut(''E''/''F''). * There is a one-to-one correspondence between subfields of ''E''/''F'' and subgroups 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「Galois extension」の詳細全文を読む スポンサード リンク
|