Galois invariants of a local field (reviewed)

Let $K^{gal}$ be the Galois closure of a $p$-adic field $K$. The first Galois invariant of $K$ is the Galois group $G=\Gal(K^{gal}/\mathbb{Q}_p)$. Most of the remaining invariants relate to the filtration on $G$ by higher ramification groups.

Let $G^i$ be the $i$th higher ramification group in upper numbering from Serre's Local Fields [10.1007/978-1-4757-5673-9]. They form a descending filtration of normal subgroups where $i$ is a continuous parameter, $i\geq -1$. Let $G^{(j)} = G^{j-1}$, and $G^{(j+)} = \cap_{\epsilon>0} G^{(j+\epsilon)}$. Then

• $G^{(0)}=G$.

• $G^{(1)}$ is the inertia subgroup of $G$, i.e., the kernel of the natural map from $G$ to the Galois group of the residue field.

• the fixed field of $G^{(1)}$ is the maximum unramified subfield of $K^{gal}/\Q_p$, which we denote by $K^{unram}$. The degree $[K^{unram}:\mathbb{Q}_p]$ is the unramified degree for $K^{gal}/\Q_p$. The extension $K^{gal}/K^{unram}$ is totally ramified.

• the group $G^{(1)}/G^{(1+)}$ is cyclic of order prime to $p$. Its order is the tame degree for $K^{gal}/\Q_p$ as it is the degree of the largest tamely totally ramified subextension.

• The group $G^{(1+)}$ is a $p$-group. Values of $s>1$ giving jumps in the filtration $G^{(s)}\neq G^{(s+)}$ are wild slopes. We repeat a slope $m$ times if $p^m =[G^{(s)} : G^{(s+)}]$, and then list the wild slopes in non-decreasing order.

• The Galois Mean Slope is the valuation of the root discriminant of the Galois closure $K^{gal}/\Q_p$. It can also be expressed as a weighted sum of the slopes.

• A Galois Splitting Model of a local field $K$ is a polynomial $f\in\Q[x]$ such that $K\cong \Q_p[x]/(f)$, and the Galois group for $f$ over $\Q_p$ equals the Galois group for $f$ over $\Q$. In this case, computations in the splitting field of $f$ over $\Q_p$ can be computed more easily using the splitting field of $f$ over $\Q$.