Corollary 1

Every finitely generated algebra $A$ over $\mathbf{Z}$ is a Jacobson ring; for a prime ideal $p \subset A$ to be maximal, it is necessary and sufficient that the ring $A/p$ be finite.

If the integral domain $A/p$ is finite, it is a field, as, for every $u \neq 0$ in $A/p$, the mapping $v \mapsto uv$ of $A/p$ to itself is injective and hence bijective since $A/p$ is finite. Conversely, for every maximal ideal $m$ of $A$, the inverse image of $m$ in $A$ is a maximal ideal ($p$) and $A/m$ is finite over the prime field $\mathbf{Z}/(p) = \mathbf{F}_p$ by Theorem 3.