On triviality of the reduced Whitehead group over Henselian fields

Abstract

Let $F$ be a Henselian field of $q$-cohomological dimension $3$, where $q$ is a prime. Let $\Gamma_F$ be the totally ordered abelian value group of $F$ and let $D$ be a central division algebra over $F$ of index a power of $q$ such that the characteristic of the residue field, $\overline{F}$ is coprime to $q$. We show that when $1\leq\dim_{\mathbb{F}_q}(\Gamma_F/q\Gamma_F)\leq 3$, the reduced Whitehead group of $D$ is trivial.