Separating transcendence basis

A transcendence basis $T$ for $L/k$ is separating if $L/k(T)$ is a separable field extension. In this case the $dt$, $t\in T$, form a basis of the Kähler differentials. In characteristic zero every transcendence basis is separating. In positive characteristic the condition on the chosen basis is essential.

For a finitely generated field extension $L/k$ in characteristic zero, a tuple $t_i$ is algebraically independent exactly when its differentials $d{t}_i$ are linearly independent in the Kähler differentials. A relation of minimum total degree gives a nontrivial differential relation; an independent tuple extends to a transcendence basis. In positive characteristic, derivatives of nonconstant relations can vanish identically.