Separably generated extension

Remarks

An algebraic extension is separably generated if and only if it is separable, since then $B = \emptyset$. In characteristic $0$ every extension is separably generated. Every finitely generated extension of a perfect field is separably generated. Example: $K(t)/K$ is separably generated, with separating transcendence basis $\set{t}$. Non-example: let $K = \F_p(t)$ and $L = K(t^{1/p})$; then $L/K$ is algebraic and purely inseparable of degree $p$, so its only transcendence basis is $\emptyset$ and $L/K$ is not separable. Not every transcendence basis of a separably generated extension is separating: $\set{t^p}$ is a transcendence basis of $\F_p(t)/\F_p$, but $\F_p(t)/\F_p(t^p)$ is purely inseparable.