- Absolute Galois group
mathematics, the absolute Galois group "GK" of a field "K" is the Galois groupof "K"sep over "K", where "K"sep is a separable closureof "K". Alternatively it is the group of all automorphisms of the algebraic closureof "K" that fix "K". The absolute Galois group is unique up toisomorphism. 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.)
* 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 is the separable closure of and .
* The absolute Galois group of a
finite field"K" is isomorphic to the group ::. The Frobenius automorphismFr is a canonical 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 Douadyand 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 Harbaterand Florian Pop, and was also proved later by Dan Haranand 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.
* No direct description is known for the absolute Galois group of the
rational numbers. In this case, it follows from Belyi's theoremthat 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 extensionof the rational numbers. Then Shafarevich's conjecture asserts that the absolute Galois group of "GK" 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 Lubotzkyand Lou van den Dries.
* M. D. Fried and M. Jarden, Field Arithmetic, Second Edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2004.
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