Interpretable Fields in Various Valued Fields

Abstract

Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ 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 $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\mathbb{Q}_p$, answering a question of Pillay's. Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if $\mathcal{K}$ is an infinite dp-minimal pure field then every field definable in $\mathcal{K}$ is definably isomorphic to a finite extension of $K$. The proof avoids elimination of imaginaries in $\mathcal{K}$ replacing it with a reduction of the problem to certain distinguished quotients of $K$.