if E is a splitting field for a division algebra D, is it always true that E can be embedded in D? Jacobson (BA II, Th. 4.8) only states that E can be embedded into Mat(r,D) for some r.



splitting field for a division algebra
It depends on what you mean by "splitting field". What is true is that for a division algebra $D$ (over $F$) of dimension $n^2$, if $E$ is a field extension of degree $n$ over $F$ that splits $D$, then $E$ is isomorphic to a subfield of $D$. See the Theorem on page 16 of Paul Garrett's [notes](http://math.umn.edu/~garrett/m/algebra/algebras.pdf).


Obviously, an algebraic closure of $F$ will split $D$, but won't be isomorphic to a subfield for dimension-counting reasons.


