(Joint work with Assaf Hasson and Kobi Peterzil)

The study of definable (or interpretable) fields in various fields has a long history, though with relatively few results, in model theory. For interpretable fields the proof usually relies on elimination of imaginaries in some well understood language.

In this talk we outline a proof that every definable field in a dp-minimal valued field K, of characteristic 0, with generic differentiability of definable functions is definably isomorphic to a finite extension of K. This latter condition holds, e.g., in p-adically closed fields, t-convex power-bounded fields and algebraically closed valued fields (really in any 1-h-minimal dp-minimal valued field).

If time permits, we will briefly outline a general method for the study of fields interpretable in dp-minimal valued fields (of characteristic 0) satisfying generic differentiability of definable functions, which bypasses elimination of imaginaries. More specifically, we show that in some situations the "interpretable" case reduces (locally) to the "definable" case.