I could not get an answer to this question in MathStackExchange, so I dare ask it here.


Given any two fields, $\rm F\_1,F\_2$ over the same prime subfield $\rm F$, the quotient $\rm \mathbf F=F\_1\otimes\_F F\_2/\mathcal M$ of the tensor product $\rm F\_1\otimes\_F F\_2$ by a maximal proper ideal $\mathcal M$ provides a field with two embeddings $$\rm F\_1\rightarrow\mathbf F,\ f\mapsto f\otimes 1 +\mathcal M\quad\text{and}\quad F\_2\rightarrow\mathbf F,\ f\mapsto 1\otimes f +\mathcal M.$$



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> **Question.** Given two division rings $\rm R\_1,R\_2$ having the same characteristic, is there a way to find a division ring $\rm R$ with two embedings $\rm R\_1⊂R$ and $\rm R\_2⊂R$ ?
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Can one embed two division rings in a common one?
Yes, this is possible. PM Cohn first showed that the amalgamated product $R\_1 \* R\_2$ over a common subfield is a "fir" (free ideal ring), and then in



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> Cohn, P.M. The embedding of firs in skewfields, Proc. London Math. Soc. (3) 23 (1971), 193–213.
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that one can adjoin inverses to get a division ring containing $R\_1 \* R\_2$.


On the other hand, the analogue is not true for division algebras (division rings which are finite-dimensional over their center). See



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> Rowen, Louis; Saltman, David. Simultaneous embeddings of finite dimensional division algebras. Proc. Amer. Math. Soc. 141 (2013), no. 3, 737–744.
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