absolute Galois group


Arithmetic geometry

Group Theory



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}.



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.



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:


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.


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?.


Of the rational numbers


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.


(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).


  • 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

category: Galois theory

Revised on February 3, 2018 16:07:49 by Ingo Blechschmidt (2a02:810d:340:511c:7dc0:662b:e5fb:92cd)