Absolute Galois group

In mathematics, the absolute Galois group "G_K" 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 algebraic 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 $Bbb\{C\}$ is the separable closure of $Bbb\{R\}$ and $[\; Bbb\{C\}\; :\; Bbb\{R\}\; ]\; =2$.
* The absolute Galois group of a finite field "K" is isomorphic to the group ::$hat\{mathbb\{Z\; =\; lim\_\{leftarrow\}mathbb\{Z\}/nmathbb\{Z\}$. The Frobenius automorphism Fr is a canonical generator of "G_K". (Recall that Fr("x") = "x^q" 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.
* Let "K" be a "p"-adic field. Then its absolute Galois group is finitely generated and has an explicit description by generators and relations.

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 "G_K" is the free profinite group of countable rank.

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 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 a absolute Galois group of a Pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries.

References

* M. D. Fried and M. Jarden, Field Arithmetic, Second Edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2004.