Dp-finite fields I: infinitesimals and positive characteristic

TL;DR

This paper establishes that NIP valued fields of positive characteristic are henselian and classifies positive characteristic dp-finite pure fields, extending known results and revealing a dichotomy in their structure.

Contribution

It proves that NIP valued fields of positive characteristic are henselian and generalizes dp-minimal field results to dp-finite fields, including a classification of such fields.

Findings

NIP valued fields of positive characteristic are henselian.

A dichotomy exists: dp-finite fields have either finite Morley rank or a non-trivial invariant valuation ring.

Positive characteristic dp-finite pure fields are classified.

Abstract

We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a field, then either K has finite Morley rank or K has a non-trivial Aut(K/A)-invariant valuation ring for a small set A. In the positive characteristic case, we can even demand that the valuation ring is henselian. Using this, we classify the positive characteristic dp-finite pure fields.