Definition 10.42.1. Let $K/k$ be a field extension.

We say $K$ is

*separably generated over $k$*if there exists a transcendence basis $\{ x_ i; i \in I\} $ of $K/k$ such that the extension $K/k(x_ i; i \in I)$ is a separable algebraic extension.We say $K$ is

*separable over $k$*if for every subextension $k \subset K' \subset K$ with $K'$ finitely generated over $k$, the extension $K'/k$ is separably generated.

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