The Stacks project

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

1. 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.

2. 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.