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absolute Galois group

Definition

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

We have $\mathrm{Gal}(K\hookrightarrow K_s)\simeq {\pi }_1(\mathrm{Spec}K)$ is equivalent to the fundamental group of the scheme $\mathrm{Spec}K$.

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

Proposition

The functor

$$\{\begin{array}{l}{\mathrm{Sch}}_{\mathrm{et}}\to \mathrm{Gal}(k\hookrightarrow k_s)-\mathrm{Set}\\ X\mapsto X(k_s)\end{array}$$\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 $G_{\mathbb{Q}}:=\mathrm{Gal}(\mathbb{Q}\hookrightarrow \overline{\mathbb{Q}})$ 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