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

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