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).

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