Let $K/k$ be a finitely generated field extension with transcendence basis $\alpha_1, \cdots \alpha_r$. I understand that this means $K$ is algebraic over $k':= k(\alpha_1, \cdots, \alpha_r)$ and $K$ being finitely generated as a $k$-algebra, we have $[K:k'] < \infty$. Now let $L$ be the separable closure of $k'$ in $K$. I have a feeling that $K$ should be purely inseparable over $L$, but I cannot prove so rigorously. Is it correct? (Please provide a proof or counterexample.)


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