Inertia group ($G_0$)

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.