Let $K^{gal}$ be the Galois closure of a local 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. They form a descending filtration of normal subgroups where $i$ is a continuous parameter, $i\geq 1$. Let $G^{(j)} = G^{j1}$, 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 nondecreasing 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 $\Q$.
 Review status: reviewed
 Last edited by John Jones on 20190503 19:35:39
 20190503 19:35:39 by John Jones (Reviewed)
 20180704 23:48:18 by John Jones (Reviewed)