Characterizing NIP henselian fields

Abstract

In this paper, we characterize NIP (Not the Independence Property) 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 nontrivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.