What is absolute galois group?

The absolute Galois group is an important concept in algebraic geometry and number theory. It is the Galois group of the algebraic closure of a field, and it plays a fundamental role in understanding the structure and properties of algebraic varieties and number fields.

The absolute Galois group is denoted by Gkbar, where k is a field. It is defined as the Galois group of the algebraic closure of k, denoted by kbar. Specifically, Gkbar is the group of all automorphisms of kbar that fix k pointwise.

The absolute Galois group is a profinite group, which means that it is a topological group that can be represented as a projective limit of finite groups. It is also a compact group, meaning that any open cover of Gkbar has a finite subcover.

One of the most important properties of the absolute Galois group is that it is a functorial object. This means that it can be used to associate a Galois group to any finite extension of k. Specifically, if L is a finite extension of k, then the Galois group of L over k is defined to be the subgroup of Gkbar that fixes L pointwise.

Another important property of the absolute Galois group is its relation to the Brauer group. The Brauer group of a field k is a group that encodes information about the central simple algebras over k. It turns out that there is a natural isomorphism between the Brauer group of k and the cohomology group H^2(Gkbar, kbar*), where kbar* is the group of units of kbar.

The absolute Galois group also has important applications in number theory. For example, it can be used to prove the famous theorem of Grothendieck that all algebraic curves over a number field k have a large "analytic" symmetry group called the etale fundamental group. This group is closely related to the absolute Galois group, and it plays a fundamental role in the study of arithmetic geometry.