Problem statement: If $K/F$ is a finite separable extension, and for any field extension $M/F$, $[M:F]$ is divisible by a fixed prime $p$, show that $[K:F]$ is a power of $p$.
Primitive element theorem tells us $K = F[\alpha]$. I am thinking about picking some element and use tower lemma, but I seem to miss pieces?
Hint: By replacing $K$ with the normal hull over $F$ we may assume that $K$ is finite and Galois over $F$. Then consider the fixed field of a Sylow $p$-subgroup of $\operatorname{Gal}(K/F)$.