Tensor product of fields and its subalgebra

Question

In Nathan Jacobson's Basic Algebra II, in section 8.18: Tensor product of fields he is discussing what happens to $E \otimes_FK$, when $K|F$ and $E|F$, and E is algebraic over F. At one point he writes "any $a \in E \otimes_FK$ is contained in a subalgebra isomorphic to an algebra $E_0 \otimes_FK$ where $E_0|F$ is finitely generated". I can't see why this is. Can anyone shed some light?

Answer 1

$a$ is a finite sum of primitive tensors $a_i\otimes b_i$, so lies in $F[a_i]\otimes K$.