On index-exponent relations over Henselian fields with local residue fields

Abstract

Let $p$ be a prime number and $(K, v)$ a Henselian valued field with a residue field $\widehat K$. This paper determines the Brauer $p$-dimension of $K$, in case $p \neq {\rm char}(\widehat K)$ and $\widehat K$ is a $p$-quasilocal field properly included in its maximal $p$-extension. When $\widehat K$ is a local field, it describes index-exponent pairs of central division $K$-algebras of $p$-primary degrees. The same goal is achieved, if $(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is local.