## primitive elements

Prove the following: Suppose $E$ is a finite separable extension of a field $F$. Then there exists $\alpha \in E$ such that $E = F(\alpha)$.

Proof. Divide in two cases: (i) $F$ is finite and (ii) $F$ is infinite. If $F$ is finite then $E$ is finite. The group of units of $E$ is cyclic. Let $\alpha$ be the generator. Then $E = F(\alpha)$.

How would you do the infinite case?