Definition 10.42.1. Let $k \subset K$ 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 $k(x_ i; i\in I) \subset K$ 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 $k \subset K'$ is separably generated.

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