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

TL;DR

This paper extends Johnson's analysis of $W_2$-topologies to fields of characteristic 2, showing that the canonical topology in unstable fields of dp-rank 2 is a $V$-topology and classifying all $W_2$-topologies in this setting.

Contribution

It generalizes Johnson's framework to characteristic 2, proving the canonical topology is a $V$-topology and classifying all $W_2$-topologies on such fields.

Findings

Canonical topology of unstable fields of characteristic 2 and dp-rank 2 is a $V$-topology.

All $W_2$-topologies on characteristic 2 fields are either from valuation rings or dense pre-diffeo-valuation data.

Completes the classification of $W_2$-topologies for all positive characteristic fields.

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.