= Solution
In characteristic $p$, let $K=\mathbb F_p(s,t)$ and $L=\mathbb F_p(s^{1/p},t^{1/p})$, with $s,t$ having <algebraic independence>. The monomials $s^{i/p}t^{j/p}$, $0\leq i,j<p$, form a $K$-basis, so $[L:K]=p^2$. Their independence follows by grouping <polynomial> monomials according to their exponents modulo $p$, after clearing rational-function denominators.
Every $\alpha\in L$ has $\alpha^p\in K$, so its <minimal polynomial> divides $T^p-\alpha^p$ and $[K(\alpha):K]\leq p$. Thus \b[this finite purely inseparable extension has no primitive element]. It demonstrates why the <separable field extension> hypothesis cannot simply be removed.
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