# Transcendental Extension is Separable over Separable Closure

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.)