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

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

How would you do the infinite case?