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For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois Theory . DEFINITION OF THE GALOIS GROUP Suppose that ''E'' is an Extension of the Field ''F''. Consider the set of all field Automorphisms of ''E''/''F''; that is, Isomorphism s α from ''E'' to itself, such that α(''x'') = ''x'' for every ''x'' in ''F''. This set of automorphisms with the operation of Function Composition forms a group ''G'', sometimes denoted Aut(''E''/''F''). If ''E''/''F'' is a Galois Extension , then ''G'' is called the Galois group of the extension, and is usually denoted Gal(''E''/''F''). EXAMPLES In the following examples ''F'' is a field, and C, '''R''', and '''Q''' are the fields of Complex , Real , and Rational numbers, respectively. The notation ''F''(''a'') indicates the Field Extension obtained by Adjoining an element ''a'' to the field ''F''.
:: :has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2.
:: : The group Aut(''K''/Q) contains only the identity automorphism. This is because ''K'' is not a Normal Extension , since the other two cube roots of 2 (both complex) are missing from the extension — in other words ''K'' is not a Splitting Field .
: :where ω is a Primitive Third Root Of Unity . The group Gal(''L''/Q) is isomorphic to S3, the Dihedral Group Of Order 6 , and ''L'' is in fact the splitting field of ''x''3 − 2 over Q. FACTS The significance of an extension being Galois is that it obeys the Fundamental Theorem Of Galois Theory : the subgroups of the Galois group correspond to the intermediate fields of the field extension. It can be shown that ''E'' is Algebraic over ''F'' if and only if the Galois group is Pro-finite . |
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