Interpretable fields in various valued fields

Abstract

Let K=(K,v,…) be a dp-minimal expansion of a non-trivially valued field of characteristic 0 and F an infinite field interpretable in K.Assume that K is one of the following: (i) V-minimal, (ii) power bounded T-convex, or (iii) P-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then F is definably isomorphic to a finite extension of K or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in Qp is definably isomorphic to a finite extension of Qp, answering a question of Pillay's.Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if K is an infinite dp-minimal pure field of characteristic 0 then every field definable in K is definably isomorphic to a finite extension of K.The proof avoids elimination of imaginaries in K replacing it with a reduction of the problem to certain distinguished quotients of K.