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

TL;DR

This paper investigates conditions under which algebraic central division algebras over Henselian fields can be decomposed into tensor products of p-primary subalgebras, linking algebraic structure to field properties.

Contribution

It establishes criteria relating the properties of Henselian fields, their residue fields, and value groups to the decomposability of central division algebras into p-primary components.

Findings

Decomposition of division algebras into tensor products of p-primary subalgebras.

Conditions on residue fields and value groups for algebra decomposability.

Existence of subalgebras matching finite-dimensional subalgebras within the decomposed structure.

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 $\widetilde R$.