# nLab absolute Galois group

###### Definition

Let $k$ be a field. Let $k_s$ denote the separable closure of $k$. Then the Galois group $Gal(k\hookrightarrow k_s)$ of the extension $k\hookrightarrow k_s$ is called absolute Galois group of $k$.

We have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the fundamental group of the scheme $Spec K$.

An instance of Grothendieck's Galois theory is the following:

###### Proposition

The functor

$\begin{cases} Sch_{et}\to Gal(k\hookrightarrow k_s)-Set \\ X\mapsto X(k_s) \end{cases}$

from the category of étale schemes to the category of sets equipped with an action of the absolute Galois group is an equivalence of categories.

###### Proposition

Recall the every profinite group appears as the Galois group of some Galois extension. Moreover we have:

Every projective profinite group appears as an absolute Galois group of a pseudo algebraically closed field?.

###### Remark

There is no direct description (for example in terms of generators and relations) known for the absolute Galois group $G_\mathbb{Q}:=Gal(\mathbb{Q}\hookrightarrow \overline \mathbb{Q})$ of the rationals.

However Belyi's theorem? implies that there is a faithful action of $G_\mathbb{Q}$ on the children's drawings.

category: Galois theory

Revised on September 12, 2012 20:36:34 by Tim Porter (95.147.237.36)