Absolute Galois group

Definition

The absolute Galois group of a field $K$ is defined as the group $\mathrm{A}\mathrm{u}\mathrm{t}(\overline{K}/K)$ where $\overline{K}$ denotes the separable closure of $K$.

The group is not just an abstract group but comes with the structure of a profinite group: it is the inverse limit of the automorphism groups for all finite Galois extensions. This also endows it with a topology (the Krull topology).