Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 3 / b

Let $G=\mathrm{Gal}(L/K)$ and normalize $v_L$. The lower ramification groups arewith $G_{-1}=G$. In particular $G_0$ is the inertia group, and $G_1$ is the wild inertia group. When the extension is totally ramified, the uniformizer criterion for lower ramification groups permits the equivalent test on one uniformizer.

LetThe extension $K_2/K$ is the unramified quadratic extension. Since ${\mathbb{F}}_{p^2}^{\times }$ contains all $(p^2-1)$st roots of unity, $L/K_2$ is a cyclic, tamely and totally ramified extension of degree $p^2-1$, with automorphisms $u\mapsto \zeta u$.

The Frobenius automorphism of ${\mathbb{F}}_{p^2}/{\mathbb{F}}_p$ extends by fixing $u$ and conjugates $\zeta$ to ${\zeta }^p$. Hence $L/K$ is Galois, its inertia group isand its residue-field Galois group isThe tame ramification also gives $G_1=1$.