Interpretable Fields in Various Valued Fields

TL;DR

This paper classifies infinite fields interpretable in certain valued fields, showing they are finite extensions of the base field or residue field, and answers a question about interpretable fields in p-adic fields.

Contribution

It provides a classification of interpretable infinite fields in dp-minimal valued fields, extending known results and avoiding elimination of imaginaries.

Findings

Any infinite interpretable field is a finite extension of the base or residue field.

Every interpretable field in dic fields is a finite extension of dic field.

The results apply to dp-minimal pure fields, showing all definable fields are finite extensions.

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 $K$.