Definable Valuations Induced by Definable Subgroups

TL;DR

This paper examines methods for defining valuations via subgroups in fields with t-henselian topologies, clarifies previous results, and corrects an earlier mistake based on detailed analysis of valuation rings induced by definable subgroups.

Contribution

It provides a detailed analysis of valuation methods used in Koenigsmann's work and corrects a key mistake in the original proof, enhancing understanding of definable valuations.

Findings

Clarification of valuation ring induction by definable subgroups

Correction of a mistake in the original valuation proof

Deeper insight into t-henselian topologies and definable valuations

Abstract

In his unpublished preprint "Definable Valuations" Koenigsmann shows that every field that admits a t-henselian topology is either real closed or separably closed or admits a definable valuation inducing the t-henselian topology. To show this Koenigsmann investigates valuation rings induced by certain (definable) subgroups of the field. The aim of this paper, based on the authors PhD thesis, is to look at the methods used in the preprint in greater detail and correct a mistake in the original paper based on a paper of Jahnke and Koenigsmann.