If $\dim_K F<\infty$, must there exist $t\in F$ such that $K[t]=F$

Question

Let $F$ be a finite dimensional extension field of a field $K$. Must there exist $t\in F$ such that $K[t]=F$?

I think I can prove this when $\operatorname{Char}K=0$, but what about the other cases?

Thank you