Let $\Delta_{1}$ and $\Delta_{2}$ be finite dimensional division algebras over field $F$ and $\Delta_{1}$ is central, then $\Delta_{1}\otimes\Delta_{2} = M_{r}(E)$ where $E$ is a division algebra, therefore we have $$\Delta_{1}^{op} \otimes \Delta_{1} \otimes \Delta_{2} \cong M_n(F) \otimes\Delta_{2} \cong M_r(E) \otimes \Delta_{1}^{op} \cong M_{r}(F) \otimes E \otimes\Delta_{1}^{op}\cong M_{r}(F) \otimes M_{q}(\Delta^{-}) \cong M_{rq}(F)\otimes\Delta^{-} $$ where n is the degree of $\Delta_{1}$ over $F$. this implies $rq=n$ and $\Delta^{-} \cong\Delta_{2}$, so $r \mid[\Delta_{1} : F]$.

MY QUESTION IS: is $r\mid[\Delta_{2} : F]$?

(so $\Delta_{1}\otimes\Delta_{2}$ will be division algebra if gcd($[\Delta_{1} : F], [\Delta_{2} : F]$) is 1)


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