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 ?
Best Answer
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.