Let $L/K$ be a field extension. How do we prove $L$ is a subfield of $M_n(K)$ (n by n matrices with entries in $K$) if and only if $[L:K]\mid n$?
My attempt: I can't prove the forward direction.
For the backward direction, ($\impliedby$)
Suppose $[L:K]=\dim_K L=d\mid n$. Then $L$ can be embedded into $M_d(K)$, which can in turn be embedded diagonally into $M_n(K)$. Thus $L$ is a subfield of $M_n(K)$.