# Finite Extension of Field of Characteristic Zero has a Primitive Element

Lemma: Let $L/K$ be a finite extension with $\text{char}(K)=0$. Then there is an $\alpha\in L$ such that $K(\alpha)=L$.

My Proof. Any extension of a field of characteristic zero is separable, so $\alpha$ exists by Existence of Primitive Element in Separable Extension.

My Problem: Isn't it true that my first sentence is only true when $L$ is either algebraic of finite? I haven't assumed either here.

Infinite separable (and thus, algebraic) extensions need not admit primitive elements, indeed, if an algebraic extension $L/K$ admits a primitive element, then it must actually be of finite degree (the degree would be equal to the degree of the minimal polynomial of the primitive element over $K$). The existence of primitive elements in separable extensions holds only for finite extensions.