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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 $M_n(K)$.

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