On algebraic central division algebras over Henselian fields of finite absolute Brauer $p$-dimensions and residually arithmetic type

Abstract

Let $(K, v)$ be a Henselian field with a residue field $\widehat K$ and value group $v(K)$, and let $\mathbb{P}$ be the set of prime numbers. This paper finds conditions on $K$, $v(K)$ and $\widehat K$ under which every algebraic associative central division $K$-algebra $R$ contains a central $K$-subalgebra $\widetilde R$ decomposable into a tensor product of central $K$-subalgebras $R _{p}$, $p \in \mathbb{P}$, of finite $p$-primary dimensions $[R _{p}\colon K]$, such that each finite-dimensional $K$-subalgebra $\Delta $ of $R$ is isomorphic to a $K$-subalgebra $\widetilde \Delta $ of $\widetilde R$.