Simultaneous Embeddings of Finite Dimensional Division Algebras

TL;DR

The paper investigates conditions under which two finite-dimensional division algebras over different fields can be embedded into a single division algebra, providing counterexamples and positive results for specific cases.

Contribution

It offers a counterexample to a general embedding question and proves positive embedding results when the division algebras are finitely generated over certain common subfields.

Findings

Counterexample to the general embedding question.

Positive embedding results for finitely generated division algebras.

Embeddings exist when the common subfield is algebraically closed or prime.

Abstract

A celebrated theorem of P.M.Cohn says that for any two division rings (not necessarily finite dimensional) over a field F, their amalgamated product over F is a domain which can be embedded in a division ring. Note that even with the two initial division rings begin finite dimensional over their centers, the resulting division ring is never finite dimensional over its center. Perhaps this led Lance Small to ask the following question. Assume $F_1$ and $F_2$ are fields with the same characteristic. Small asked whether any two division algebras $D_1/F_1$ and $D_2/F_2$ can be embedded in some third division algebra $E/F$. We start with a surprisingly straightforward counterexample, but then show that a positive solution exists for division algebras finitely generated over a common subfield which is either algebraically closed or the prime subfield.