Unramified extension

A finite extension of local fields is unramified when its ramification index is one and its residue-field degree equals the field degree. A local field has a unique unramified extension of each positive degree inside a fixed algebraic closure.

For a finite Galois extension of local fields, the lower ramification groups measure how closely automorphisms fix the valuation ring. The inertia group is $G_0$, and $G_1$ is the wild inertia group.

The inertia group of a finite Galois extension of local fields is the kernel of the action of its Galois group on the residue field. In lower numbering it is the zeroth ramification group $G_0$.

The wild inertia group is the first ramification group $G_1$. It is the unique Sylow subgroup of the inertia group for the residue characteristic.