Characterizing NIP henselian fields

TL;DR

This paper provides a model-theoretic and algebraic characterization of NIP henselian valued fields, advancing the understanding of their structure and contributing to the broader classification of NIP fields under certain conjectures.

Contribution

It offers a new characterization of NIP henselian valued fields modulo residue field theory, assuming a conjecture about infinite NIP fields, thus advancing classification efforts.

Findings

Characterization of NIP henselian valued fields in algebraic and model-theoretic terms

Conditional classification of all NIP fields based on a conjecture

Progress towards understanding the structure of NIP fields

Abstract

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.