The canonical topology on dp-minimal fields

Abstract

We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].