Fields of dp-Rank 2 and their W_2-Topologies: The Characteristic 2 Case

Abstract

In this note we reproduce Johnson's analysis of $W_2$-topologies on fields of characteristic 2, which was originally stated for fields of characteristic different than 2. Following his framework, we prove that the canonical topology of an unstable field of characteristic 2 and dp-rank 2 is a $V$-topology. Additionally, we show that any $W_2$-topology on a field of characteristic 2 is either induced by the intersection of two valuation rings or it is induced by dense pre-diffeo-valuation data, completing the picture for all positive characteristic fields.