# For $K/F$ be a separably generated field extension and for any finitely generated intermediate field $L$ $L/F$ is also separably generated.

Let $$K/F$$ be a separably generated field extension. Then how can I show that every intermediate field $$L$$ with $$L$$ finitely generated over $$F$$ the extension $$L/F$$ is also separably generated.

If there doesn't exist any separating transcendence base for $$L/F$$ there is always some purely inseparable portion of $$L$$ over the field generated by a(any) transcendence base of $$L/F$$ over $$F$$. But I can't draw any contradiction from here. How can I show this ?

An extension $$K/F$$ is separable if and only if $$M\otimes_FK$$ is reduced for all field extensions $$M/F$$. If $$L$$ is an intermediate field of $$K/F$$, then $$M\otimes_FL$$ is a subalgebra of $$M\otimes_FK$$. Finally, subalgebras of reduced algebras are themselves reduced.
Putting this together shows that $$K/F$$ separable implies $$L/F$$ separable.
• I think it is worth mentioning (unless you have something else in mind) that to say $M\otimes _F L$ is a subalgebra of $M\otimes_F K$ uses the flatness of $M$ as an $F$-module.