In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line:
Since the field $\mathbb Q (a_1,\ldots, a_l)$ is finitely generated, the field $K$ of algebraic numbers in it is finite over $\mathbb Q$.
Why is this so? I can't find a proof for it anywhere. My best guess is that $K$ is a finitely generated algebra over $\mathbb Q$, and hence by the weak Nullstellensatz is of finite degree, but I can't see why $K$ would be finitely generated.