Characterising local fields of positive characteristic by Galois theory and the Brauer group

TL;DR

This paper characterizes local fields of positive characteristic using their Galois groups and Brauer groups, providing a new way to distinguish these fields within a broader class of imperfect fields.

Contribution

It introduces a novel characterization of local fields of characteristic p via Galois groups and Brauer group properties, extending previous work on Galois group isomorphisms.

Findings

Local fields of characteristic p are uniquely characterized by their Galois groups and Brauer group properties.

The work complements prior analyses by Efrat-Fesenko on Galois group isomorphisms.

Provides a new framework for identifying local fields within imperfect fields.

Abstract

We show that each local field $\mathbb{F}_q((t))$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most $1$ by (certain small quotients of) its absolute Galois group together with natural axioms concerning the $p$-torsion of its Brauer group. This complements previous work by Efrat-Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic $p$.