# Simultaneous Embeddings of Finite Dimensional Division Algebras - Mathematics > Rings and Algebras

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Abstract: A celebrated theorem of P.M.Cohn says that for any two division rings notnecessarily finite dimensional over a field F, their amalgamated product overF is a domain which can be embedded in a division ring. Note that even with thetwo initial division rings begin finite dimensional over their centers, theresulting division ring is never finite dimensional over its center. Perhapsthis led Lance Small to ask the following question. Assume $F 1$ and $F 2$ arefields with the same characteristic. Small asked whether any two divisionalgebras $D 1-F 1$ and $D 2-F 2$ can be embedded in some third division algebra$E-F$. We start with a surprisingly straightforward counterexample, but thenshow that a positive solution exists for division algebras finitely generatedover a common subfield which is either algebraically closed or the primesubfield.

Author: ** Louis Rowen, David J Saltman**

Source: https://arxiv.org/