On dp-minimal fields

TL;DR

This paper classifies dp-minimal pure fields, showing most are equivalent to Hahn series fields with specific properties, and establishes that dp-small fields are algebraically or real closed.

Contribution

It provides a classification of dp-minimal pure fields and characterizes dp-small fields as algebraically closed or real closed.

Findings

Most dp-minimal fields are equivalent to certain Hahn series fields.

Dp-small fields, including VC-minimal fields, are algebraically closed or real closed.

The classification depends on divisibility conditions and base fields like _p^{alg} or local fields.

Abstract

We classify dp-minimal pure fields up to elementary equivalence. Most are equivalent to Hahn series fields $K((t^\Gamma))$ where $\Gamma$ satisfies some divisibility conditions and $K$ is $\mathbb{F}_p^{alg}$ or a local field of characteristic zero. We show that dp-small fields (including VC-minimal fields) are algebraically closed or real closed.