transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
The absolute Galois group of a field is that of the field extension which is the separable closure of . When is a perfect field this is equivalently the Galois group of the algebraic closure .
Let be a field. Let denote the separable closure of . Then the Galois group of the field extension is called absolute Galois group of .
By general Galois theory we have is equivalent to the fundamental group of the spectrum scheme
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 of the rational numbers (with being the algebraic numbers).
However Belyi's theorem? implies that there is a faithful action of 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
Last revised on September 25, 2018 at 15:26:41. See the history of this page for a list of all contributions to it.