The classification of dp-minimal and dp-small fields

Abstract

Continuing the work of Johnson (2018), we classify the pure fields and pure valued fields of dp-rank 1, up to elementary equivalence. In the process, we give a few new examples of dp-minimal fields. Specializing to the case of dp-small fields, we prove that dp-small fields and VC-minimal fields are all algebraically closed or real closed, as had been conjectured by Guingona (2014). This generalizes earlier work of Haskell and Macpherson (1994) and Macpherson et al. (2000), showing that $C$-minimal fields are algebraically closed, and weakly o-minimal fields are real closed. In fact, we obtain slight strengthenings of these earlier results, since we assume no compatibility between the field structure on the one hand and the $C$-relation or order on the other hand. We also give a new proof of Jahnke, Simon, and Walsberg’s (2017) theorem that dp-minimal valued fields are henselian. Lastly, we apply the Kaplan–Scanlon–Wagner (2011) theorem (NIP fields are Artin–Schreier closed) to prove some general properties of strongly dependent valued fields. For example, strongly dependent henselian valued fields are defectless.