# Absolute Galois group The absolute Galois group of the real numbers is a cyclic group of order 2 generated by complex conjugation, since C is the separable closure of R and [C:R] = 2.

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep 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 well-defined up to inner automorphism. It is a profinite group.

(When K is a perfect field, Ksep is the same as an algebraic closure Kalg 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 [C:R] = 2.
• The absolute Galois group of a finite field K is isomorphic to the group
${\hat {\mathbf {Z} }}=\varprojlim \mathbf {Z} /n\mathbf {Z} .$ (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 Kalg, 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 Qp. For p ≠ 2, its absolute Galois group is generated by [K:Qp] + 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.

## Problems

• No direct description is known for the absolute Galois group of the rational numbers. 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.