Let $L/K$ be a finite separable extension of fields. Then we have the primitive element theorem, i.e., there exists an $x$ in $L$ such that $L=K(x)$.
In particular, the primitive element theorem holds for all finite extensions of a perfect field.
Question 1. Is a field $K$ perfect if and only if the primitive element theorem holds for all finite extensions of $K$?
Question 2. Suppose that $K$ is a field extension of $\mathbf{F}_p$ of transcendence degree 1, i.e., a function field over a finite field. Does the primitive element theorem hold for any finite extension of $K$?
In Question 2, I am actually only interested in the case $K=\mathbf{F}_q(t)$.