Absolute Galois group

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In mathematics, the absolute Galois group of a field K is the Galois group of K^sep over K, where K^sep is a separable closure of K. When K is a perfect field, K^sep is the same as an algebraic closure of K, and therefore this definition may be used for K of characteristic zero, or a finite field. The absolute Galois group is unique up to isomorphism. It is a profinite group.

Absolute Galois groups are hard to describe directly, for example for K 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'enfant of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.