NIP henselian valued fields

TL;DR

This paper characterizes when henselian valued fields are NIP based on properties of their residue fields and value groups, providing a criterion that links model-theoretic tameness to algebraic conditions.

Contribution

It establishes a precise equivalence for NIP in tame henselian valued fields, connecting the NIP property of the field with that of its residue field and value group, and introduces the concept of roughly tame fields.

Findings

NIP of tame henselian valued fields is characterized by NIP of residue field and value group.

Henselian valued fields with finite $K^ imes/(K^ imes)^p$ are NIP iff residue field is NIP and the field is roughly tame.

Provides a criterion for NIP in fields with residue characteristic p, linking algebraic properties to model-theoretic tameness.

Abstract

We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if $(K,v)$ is a henselian valued field of residue characteristic $\mathrm{char}(Kv)=p$ for which $K^\times/(K^\times)^p$ is finite in case $p>0$, then $(K,v)$ is NIP iff $Kv$ is NIP and $v$ is roughly tame.