Definition 1

Let $L$ be an extension of a field $K$. A transcendence basis $B$ of $L$ over $K$ is said to be separating if the algebraic extension $L$ of $K(B)$ is separable.

If $K$ has characteristic 0, every transcendence basis of $L$ over $K$ is separating because every algebraic extension of a field of characteristic 0 is separable ($V$, p. 37, Cor.). If an extension admits a separating transcendence basis, it is separable ($V$, p. 121, Prop. 6 and p. 122, Prop. 9). The following theorem shows that every finitely generated separable extension admits a separating transcendence basis; this restriction is essential ($V$, p. 177, Ex. 1).