On the Brauer $p$-dimension of Henselian discrete valued fields of residual characteristic $p > 0$

TL;DR

This paper investigates the Brauer p-dimension of Henselian discrete valued fields with residue characteristic p, establishing a direct link between the dimension and the degree of the residue field extension.

Contribution

It provides a precise characterization of the Brauer p-dimension in terms of the residue field extension degree, including conditions for it to be infinite.

Findings

Brd_p(K) ≥ n if [\widehat{K} : \\widehat{K}^p] = p^n

Brd_p(K) = ∞ iff [\widehat{K} : \\widehat{K}^p] = ∞

Establishes a criterion connecting residue field extension degree with Brauer p-dimension

Abstract

Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $p$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$. This paper shows that Brd$_{p}(K) \ge n$, if $[\widehat K\colon \widehat K ^{p}] = p ^{n}$, for some $n \in \mathbb{N}$. It proves that Brd$_{p}(K) = \infty $ if and only if $[\widehat K\colon \widehat K ^{p}] = \infty $.