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absolute Galois group

Contents

Idea

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

Definition

Definition

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

Properties

Remark

By general Galois theory we have Gal(KK s)π 1(SpecK)Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K) is equivalent to the fundamental group of the spectrum scheme SpecKSpec K

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

Proposition

The functor

{Sch etGal(kk s)Set XX(k s)\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 :=Gal(¯)G_\mathbb{Q}:=Gal(\mathbb{Q}\hookrightarrow \overline \mathbb{Q}) of the rational numbers.

However Belyi's theorem? implies that there is a faithful action of G 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

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

category: Galois theory

Revised on June 16, 2014 04:51:00 by Urs Schreiber (88.128.80.80)