Commuting-Liftable Subgroups of Galois Groups II

Adam Topaz

Published 2012-08-02, updated 2015-01-23Version 5

Let $n$ denote either a positive integer or $\infty$, let $\ell$ be a fixed prime and let $K$ be a field of characteristic different from $\ell$. In the presence of sufficiently many roots of unity in $K$, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal $\ell^n$-abelian Galois group of $K$ using the maximal $\ell^N$-abelian-by-central Galois group of $K$, whenever $N$ is sufficiently large relative to $n$.