absolute Galois group

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:

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.

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?.

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)