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In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is unique up to isomorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) == Examples == * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and () = 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group :: (For the notation, see Inverse limit.) :The Frobenius automorphism Fr is a canonical (topological) generator of ''GK''. (Recall that Fr(''x'') = ''xq'' for all ''x'' in ''K''alg, where ''q'' is the number of elements in ''K''.) * The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem. * More generally, let ''C'' be an algebraically closed field and ''x'' a variable. Then the absolute Galois group of ''K'' = ''C''(''x'') is free of rank equal to the cardinality of ''C''. This result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden using algebraic methods. * Let ''K'' be a finite extension of the p-adic numbers Q''p''. For ''p'' ≠ 2, its absolute Galois group is generated by () + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case ''p'' = 2, but the structure for Q2 is not known. *Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers.〔http://math.uci.edu/~mfried/paplist-cov/QTotallyReal.pdf〕 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「Absolute Galois group」の詳細全文を読む スポンサード リンク
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