# Definable Valuations induced by multiplicative subgroups and NIP Fields

## TL;DR

This paper investigates how NIP and dp-minimality properties influence the algebraic structure of infinite fields, especially Hahn fields, supporting the conjecture that certain NIP fields admit definable henselian valuations.

## Contribution

It advances understanding of definable valuations in NIP fields, particularly in Hahn fields, and builds on prior work on dp-minimal fields.

## Key findings

- NIP fields with certain properties admit definable henselian valuations
- Results support the conjecture for fields not real closed or separably closed
- Focus on algebraic implications in Hahn fields

## Abstract

We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a definable henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015.