Commuting-Liftable Subgroups of Galois Groups II

TL;DR

This paper explores how to recover valuation inertia and decomposition structures within Galois groups of fields using abelian-by-central Galois groups, under certain conditions involving roots of unity and large enough N.

Contribution

It introduces a method to recover valuation structures from Galois groups using abelian-by-central quotients, extending previous work to a broader setting.

Findings

Recovery of valuation inertia/decomposition structures from Galois groups

Conditions involving roots of unity are crucial for the method

Applicable for sufficiently large N relative to n

Abstract

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 $n$.