Henselian discrete valued stable fields

TL;DR

This paper characterizes the Brauer p-dimension of Henselian discrete valued fields based on properties of their residue fields, revealing conditions for when this dimension is at most one.

Contribution

It provides a complete characterization of the Brauer p-dimension for Henselian discrete valued fields in relation to their residue fields, extending previous results.

Findings

Brd_p(K) ≤ 1 iff residue field is p-quasilocal and degree [: ^p] e0 p when p=q.

For p e0 q, Brd_p(K) e0 1 iff residue field is p-quasilocal and degree e0 p.

Results extend understanding of Brauer dimensions in valued fields.

Abstract

Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $q \ge 0$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$, for each prime $p$. The present paper shows that if $p = q$, then Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is a $p$-quasilocal field and the degree $[\widehat K\colon \widehat K ^{p}]$ is $\le p$. This complements our earlier result that, in case $p \neq q$, we have Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is $p$-quasilocal and Brd$_{p}(\widehat K) \le 1$.