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In [[mathematics]], the '''absolute Galois group''' ''GK'' of a [[field (mathematics)|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 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.
* The absolute Galois group of a [[finite field]] ''K'' is isomorphic to the group
::<math> \hat{\mathbf{Z}} = \varprojlim \mathbf{Z}/n\mathbf{Z}. </math>
(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]].<ref>{{harvnb|Douady|1964}}</ref>
* 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>
* Let ''K'' be a [[finite extension]] of the [[p-adic number]]s '''Q'''''p''. For ''p'' ≠ 2, its absolute Galois group is generated by [''K'':'''Q'''''p''] + 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'''2 is not known.<ref>{{harvnb|Neukirch|Schmidt|Wingberg|2000|loc=§VII.5}}</ref>
*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>

== Problems ==

* 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.

* 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>

== Some general results ==
* 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.

* 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>

== Notes ==
{{reflist}}

== References ==

*{{Citation
| last=Douady
| first=Adrien
| title=Détermination d'un groupe de Galois
| year=1964
| mr=0162796
| journal=Comptes Rendues de l'Académie des Sciences de Paris
| volume=258
| pages=5305–5308
}}
* {{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 }}
*{{Citation
| last=Haran
| first=Dan
| last2=Jarden
| first2=Moshe
| title=The absolute Galois group of ''C''(''x'')
| journal=Pacific Journal of Mathematics
| year=2000
| volume=196
| issue=2
| mr=1800587
| pages=445–459
}}
*{{Citation
| last=Harbater
| first=David
| author-link=David Harbater
| contribution=Fundamental groups and embedding problems in characteristic ''p''
| mr=1352282
| pages=353–369
| title=Recent developments in the inverse Galois problem
| publisher=[[American Mathematical Society]]
| location=[[Providence, RI]]
| series=Contemporary Mathematics
| volume=186
}}
*{{Citation
| last=Jannsen
| first=Uwe
| last2=Wingberg
| first2=Kay
| title=Die Struktur der absoluten Galoisgruppe <math>\mathfrak{p}</math>-adischer Zahlkörper
| journal=[[Inventiones Mathematicae]]
| volume=70
| year=1982
| pages=71–78
}}
*{{Neukirch et al. CNF|edition=1}}
*{{Citation
| last=Pop
| first=Florian
| author-link=Florian Pop
| title=Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture
| journal=[[Inventiones Mathematicae]]
| volume=120
| issue=3
| year=1995
| pages=555–578
| mr=1334484
}}

[[Category:Galois theory]]

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Cerebral perfusion pressure