 Purely inseparable extension without a primitive element

Let $K={\mathbb{F}}_p(s,t)$ and $L=K(s^{1/p},t^{1/p})$ for algebraically independent $s,t$. The $p^2$ monomials $s^{i/p}{t}^{j/p}$ with $0\le i,j<p$ form a basis, whereas every $\alpha \in L$ satisfies ${\alpha }^p\in K$. Thus $[L:K]=p^2$ but $[K(\alpha ):K]\le p$, so no primitive element of a field extension exists.