A global function field is a finite extension of $\mathbb{F}_p(T)$ for some prime number $p$. A finite field extension $L/K$ is called simple if it has a primitive element, i.e. there exists an $x \in L$ such $L$ is generated (as a $K$-algebra) by $x$.
Finite extensions of global functions fields definitely do not need to be separable, but are they always simple?
