Abstract

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