If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field.

Question

I saw in Jacobson's book Finite Dimensional Division Algebras (1996) page 158 the sentence "If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field." Here A is an F-central simple algebra. I don't understand how to prove it.

Answer 1

$E$ being a splitting field means that we have an isomorphism

$$\varphi : M_n(E) \cong A \otimes_F E$$

for some $n$. The point now is that only finitely many elements of $E$ participate in this isomorphism. Namely, consider the standard basis $e_{ij} \in M_n(E)$. $\varphi$ is completely determined by the images $\varphi(e_{ij}) \in A \otimes_F E$, and each of these is a finite sum of elements in $A \otimes_F E$. Let $E'$ be the subfield of $E$ generated over $F$ by these elements; then $\varphi(e_{ij}) \in A \otimes_F E'$ for all $i, j$ (by construction), hence the isomorphism above is actually defined over $E'$.