I have it in my lectures notes that the claim: Let $K/F$ be a finite and separable extension then $K$ is a simple extension of $F$ follows immediately from the theorom : Let $K/F$ be a finite extension, then it is simple iff $K/F$ have a finite number of subfiels.
My question is why ? I know that the extension is finite and separable hence $K=F(a_1,...a_n)$ where $a_i$ are all separable, but why there are only finite number of subfields between $F$ and $K$ ?