Purely inseparable extension without a primitive element
= 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 $0\leq i,j<p$ form a basis, whereas every $\alpha\in L$ satisfies $\alpha^p\in K$. Thus $[L:K]=p^2$ but $[K(\alpha):K]\leq p$, so no <primitive element of a field extension> exists.