I have a proof of the primitive element theorem for subfields $K$ of $\mathbb{C}$ which relies on there being infinitely many elements in $K$. In a book I've been reading it states that every finite separable extension is simple and that this follows from the earlier version the primitive element theorem.
Surely this doesn't work if I'm in a finite field? Is there another proof of the primitive element theorem that works in this case? Or is there a more general version that doesn't required infinitely many elements in $K$?