Abelian-by-Central Galois Groups of Fields II: Definability of Inertia/Decomposition Groups

TL;DR

This paper demonstrates that minimized inertia and decomposition groups in certain Galois groups are definable using first-order logic, advancing understanding of valuation theory in field extensions.

Contribution

It proves the first-order definability of inertia and decomposition groups in pro- abelian-by-central Galois groups using commuting-liftable pairs.

Findings

Minimized inertia groups are first-order definable in many valuations.

Minimized decomposition groups are also first-order definable.

The approach uses properties of commuting-liftable pairs in Galois groups.

Abstract

This paper explores some first-order properties of commuting-liftable pairs in pro-$\ell$ abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs.