Absolute Galois group

In mathematics, particularly in anabelian geometry and p-adic geometry, the absolute Galois group ${\displaystyle G_{K}}$ of a field ${\displaystyle K}$ is the Galois group of ${\displaystyle K^{\textrm {sep}}}$ over ${\displaystyle K}$, where ${\displaystyle K^{\textrm {sep}}}$ is a separable closure of ${\displaystyle K}$. Alternatively, it is the group of all automorphisms of the algebraic closure of ${\displaystyle K}$ that fix ${\displaystyle K}$. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.

When ${\displaystyle K}$ is a perfect field, ${\displaystyle K^{\textrm {sep}}}$ is the same as an algebraic closure ${\displaystyle K^{\textrm {alg}}}$ of ${\displaystyle K}$. Notably, this holds when ${\displaystyle K}$ has characteristic zero or is finite.