Henselian discrete valued stable fields

Abstract

Let $(K,v)$ be a Henselian discrete valued field with residue field $\hat{K}$ 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 $\hat{K}$ is a $p$-quasilocal field and the degree $[\hat{K}:{\hat{K}}^p]$ is $\le p$. This complements our earlier result that, in case $p\ne q$, we have Brd${}_p(K)\le 1$ if and only if $\hat{K}$ is $p$-quasilocal and Brd${}_p(\hat{K})\le 1$.

15 pages, LaTeX. Accepted for publication in a Special Issue of Turk. J. Math., dedicated to Vesselin Drensky on the occasion of his 70th anniversary