Absolute Galois group

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In mathematics, the absolute Galois group of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. The absolute Galois group is unique up to isomorphism. 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.)

[edit] Examples

  • 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 is isomorphic to the group \hat{\mathbb{Z}} = \lim_{\leftarrow}\mathbb{Z}/n\mathbb{Z} . This is shown by Galois theory.
  • 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.