Embedding of central simple algebras A paper I am reading has the following statement (without proof):
If $B$ and $A$ are central simple algebras with the same center $K$
where $K$ is a local or a global field, then $B$ can be embedded into $A$
if and only if [B : K] is relatively prime with $[A : K]/[B : K]$.
The necessity is fairly easy to show. But how does one show the sufficiency? 

This sounds very unlikely to me. What if $A$ is a division algebra
and $B$ is a matrix algebra over $K$? (When $K=\Bbb Q_p$, theses exist with any given square degree over $K$.)
The other way round sounds fishy too. Over $K=\Bbb Q_p$ there are
division algebras $A$ and $B$ with $|A:K|=16$ and $|B:K|=4$
and with $B$ a subalgebra of $A$.
