Consider $M_n(F)$ for a field $F$, and suppose $K\subseteq M_n(F)$ is isomorphic to an extension field of $F$. Is it true that $[K:F]\leq n$?
If $K$ is separable, then the primitive element theory gives that $K=F(\alpha)$, and so $\alpha \in M_n(F)$ will have a minimal polynomial of degree at most $n$, so $[K:F]\leq n$.
But if $K$ is not separable, that logic does not work. The primitive element theorem holds more generally if the set of subfields $F\subseteq L\subseteq K$ is finite, but I don't see how I could deduce that from $K$ being a isomorphic to part of a matrix algebra.