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?