Given a field $F$ and a field extension $K$ such that $K$ is a finitely generated $F$-algebra, show that $K$ is algebraic over $F$.
I think the result should follow by induction on the cardinality of a minimal generating set for $K$ as an $F$-algebra, but I haven't been able to proceed with that line of argument. I am guessing one needs to use some results about integral extensions, but I am not sure.