Contents

group theory

# Contents

## Idea

The absolute Galois group of a field $k$ is that of the field extension $k \hookrightarrow k_s$ which is the separable closure of $k$. When $k$ is a perfect field this is equivalently the Galois group of the algebraic closure $k \hookrightarrow \overline{k}$.

## Definition

###### 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 field extension $k\hookrightarrow k_s$ is called absolute Galois group of $k$.

## Properties

###### Remark

By general Galois theory we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the fundamental group of the spectrum 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?.

## Examples

### Of the rational numbers

###### Remark

There is no direct description (for example in terms of generators and relations) known for the absolute Galois group $G_\mathbb{Q} \coloneqq Gal(\mathbb{Q}\hookrightarrow \overline{\mathbb{Q}})$ of the rational numbers (with $\overline{\mathbb{Q}}$ being the algebraic numbers).

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

###### Theorem

(Drinfeld, Ihara, Deligne)

There is an inclusion of the absolute Galois group of the rational numbers into the Grothendieck-Teichmüller group (recalled e.g. as Stix 04, theorem 6).

## References

### General

• Jakob Stix, The Grothendieck-Teichmüller group and Galois theory of the rational numbers, 2004 (pdf)

Discussion of the p-adic absolute Galois group as the etale fundamental group of a quotient of some perfectoid space is in