Henselian expansions of NIP fields

Abstract

Let [Formula: see text] be an NIP field and let [Formula: see text] be a Henselian valuation on [Formula: see text]. We ask whether [Formula: see text] is NIP as a valued field. By a result of Shelah, we know that if [Formula: see text] is externally definable, then [Formula: see text] is NIP. Using the definability of the canonical [Formula: see text]-Henselian valuation, we show that whenever the residue field of [Formula: see text] is not separably closed, then [Formula: see text] is externally definable. In the case of separably closed residue field, we show that [Formula: see text] is NIP as a pure valued field.