Remark

— Let $L$ be a separable finitely generated extension of a field $K$ of characteristic $p \neq 0$; there may exist transcendence bases of $L$ which are not separating. It is enough to observe that $\{X^p\}$ is a transcendence basis of $K(X)$ but $K(X)$ is a p-radical extension of degree $p$ of $K(X^p)$.