Submission date: 30 April 2020

Abstract:

We prove that unstable dp-finite fields admit definable V-topologies. As a consequence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof of the classification of dp-finite fields of positive characteristic. For n ≥ 1, we define a local class of “W_n-topological fields”, generalizing V-topological fields. A W_1-topology is the same thing as a V-topology, and a W_n-topology is some higher-rank analogue. If K is an unstable dp-finite field, then the canonical topology is a definable W_n-topology for n = dp-rk(K). Every W_n-topology has between 1 and n coarsenings that are V-topologies. If the given W_n-topology is definable in some structure, then so are the V-topological coarsenings.

Mathematics Subject Classification: 03C45 (Primary) 12J99 (Secondary)

Keywords and phrases:

Full text arXiv 2004.14732: pdf, ps.