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| In [[mathematics]], the '''absolute Galois group''' ''G<sub>K</sub>'' of a [[field (mathematics)|field]] ''K'' is the [[Galois group]] of ''K''<sup>sep</sup> over ''K'', where ''K''<sup>sep</sup> 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]].
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| (When ''K'' is a [[perfect field]], ''K''<sup>sep</sup> is the same as an [[algebraic closure]] ''K''<sup>alg</sup> of ''K''. This holds e.g. for ''K'' of [[characteristic zero]], or ''K'' a [[finite field]].)
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| == Examples ==
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| * The absolute Galois group of an algebraically closed field is trivial.
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| * The absolute Galois group of the [[real number]]s is a cyclic group of two elements (complex conjugation and the identity map), since '''C''' is the separable closure of '''R''' and ['''C''':'''R'''] = 2.
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| * The absolute Galois group of a [[finite field]] ''K'' is isomorphic to the group
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| ::<math> \hat{\mathbf{Z}} = \varprojlim \mathbf{Z}/n\mathbf{Z}. </math>
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| (For the notation, see [[Inverse limit]].)
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| :The [[Frobenius automorphism]] Fr is a canonical (topological) generator of ''G<sub>K</sub>''. (Recall that Fr(''x'') = ''x<sup>q</sup>'' for all ''x'' in ''K''<sup>alg</sup>, where ''q'' is the number of elements in ''K''.)
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| * 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]].<ref>{{harvnb|Douady|1964}}</ref>
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| * 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.<ref>{{harvnb|Harbater|1995}}</ref><ref>{{harvnb|Pop|1995}}</ref><ref>{{harvnb|Haran|Jarden|2000}}</ref>
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| * Let ''K'' be a [[finite extension]] of the [[p-adic number]]s '''Q'''<sub>''p''</sub>. For ''p'' ≠ 2, its absolute Galois group is generated by [''K'':'''Q'''<sub>''p''</sub>] + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg.<ref>{{harvnb|Jannsen|Wingberg|1982}}</ref><ref>{{harvnb|Neukirch|Schmidt|Wingberg|2000|loc=theorem 7.5.10}}</ref> Some results are known in the case ''p'' = 2, but the structure for '''Q'''<sub>2</sub> is not known.<ref>{{harvnb|Neukirch|Schmidt|Wingberg|2000|loc=§VII.5}}</ref>
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| *Another case in which the absolute Galois group has been determined is for the largest [[totally real]] subfield of the field of algebraic numbers.<ref>http://math.uci.edu/~mfried/paplist-cov/QTotallyReal.pdf</ref>
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| == Problems ==
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| * No direct description is known for the absolute Galois group of the [[rational number]]s. In this case, it follows from [[Belyi's theorem]] that the absolute Galois group has a faithful action on the ''[[dessins d'enfants]]'' of [[Grothendieck]] (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.
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| * Let ''K'' be the maximal [[abelian extension]] of the rational numbers. Then '''Shafarevich's conjecture''' asserts that the absolute Galois group of ''K'' is a free profinite group.<ref>{{harvnb|Neukirch|Schmidt|Wingberg|2000}}, p. 449.</ref>
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| == Some general results ==
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| * Every profinite group occurs as a Galois group of some Galois extension, however not every profinite group occurs as an absolute Galois group. For example, the [[Real closed field|Artin–Schreier theorem]] asserts that the only finite absolute Galois groups are the trivial one and the cyclic group of order 2.
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| * Every [[projective profinite group]] can be realized as an absolute Galois group of a [[pseudo algebraically closed field]]. This result is due to [[Alexander Lubotzky]] and [[Lou van den Dries]].<ref>Fried & Jarden (2008) pp.208,545</ref>
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| == Notes ==
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| {{reflist}}
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| == References ==
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| *{{Citation
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| | last=Douady
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| | first=Adrien
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| | title=Détermination d'un groupe de Galois
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| | year=1964
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| | mr=0162796
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| | journal=Comptes Rendues de l'Académie des Sciences de Paris
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| | volume=258
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| | pages=5305–5308
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| }}
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| * {{citation | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=2nd revised and enlarged | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2004 | isbn=3-540-22811-X | zbl=1055.12003 }}
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| *{{Citation
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| | last=Haran
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| | first=Dan
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| | last2=Jarden
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| | first2=Moshe
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| | title=The absolute Galois group of ''C''(''x'')
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| | journal=Pacific Journal of Mathematics
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| | year=2000
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| | volume=196
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| | issue=2
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| | mr=1800587
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| | pages=445–459
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| }}
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| *{{Citation
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| | last=Harbater
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| | first=David
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| | author-link=David Harbater
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| | contribution=Fundamental groups and embedding problems in characteristic ''p''
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| | mr=1352282
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| | pages=353–369
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| | title=Recent developments in the inverse Galois problem
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| | publisher=[[American Mathematical Society]]
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| | location=[[Providence, RI]]
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| | series=Contemporary Mathematics
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| | volume=186
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| }}
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| *{{Citation
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| | last=Jannsen
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| | first=Uwe
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| | last2=Wingberg
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| | first2=Kay
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| | title=Die Struktur der absoluten Galoisgruppe <math>\mathfrak{p}</math>-adischer Zahlkörper
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| | journal=[[Inventiones Mathematicae]]
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| | volume=70
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| | year=1982
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| | pages=71–78
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| }}
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| *{{Neukirch et al. CNF|edition=1}}
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| *{{Citation
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| | last=Pop
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| | first=Florian
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| | author-link=Florian Pop
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| | title=Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture
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| | journal=[[Inventiones Mathematicae]]
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| | volume=120
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| | issue=3
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| | year=1995
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| | pages=555–578
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| | mr=1334484
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| }}
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| [[Category:Galois theory]]
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