transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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 $k \hookrightarrow \overline{k}$.
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 $k$.
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:
The functor
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?.
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.
(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).
Discussion of the p-adic absolute Galois group as the etale fundamental group of a quotient of some perfectoid space is in
See also
Discussion in the context of string theory includes