On triviality of 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 $1\le {\dim }_{{\mathbb{F}}_q}({\Gamma }_F/q{\Gamma }_F)\le 3$, the reduced Whitehead group of $D$ is trivial.

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• Trivialité des groupes de Whitehead réduits avec applications à l'approximation faible et l'approximation forte