Let $F$ be a field, and $K$ a field extension of $F$. Prove that if $[K:F]$ is prime, then there is no field $L$ such that $F\subset L\subset K$ and $F\neq L,L\neq K$.
Well, if $L$ is an extension of $F$ and $K$ is an extension of $L$, then we can say $[K:F]=[K:L][L:F]$, yielding that either $K=L$ or $L=F$.
But here $L$ is any field between $F$ and $K$. What should we do?