Proof on simple extension

Question

Let $E$ be a finite extension of $F$ and suppose that $[E:F] = p$ where $p$ is prime. Prove that $E$ is a simple extension of $F$.

I really don't know where to start. Thanks in advance.

Answer 1

Suppose, $\ G\ $ is a field with $\ F\le G\le E\ $. Then, we have $$[E:F]=[E:G]\cdot [G:F]$$ Since $\ [E:F]\ $ is prime, we get $\ E=G\ $ or $\ G=F\ $. Hence $\ G\ $ cannot be a proper intermediate field, hence the extension must be simple.