Property of prime ideals of $\Bbb{Z}[X_1,...,X_n]$

Question

Let $P$ be a prime ideal of $\Bbb{Z}[X_1,...,X_n]$. How to show that there exist a prime number $p$ such that $(p)+P$ is not $\Bbb{Z}[X_1,...,X_n]$.

Answer 1

By considering $A := \mathbb{Z}[X_1,\dotsc,X_n]/P$, the question is equivalent to: Let $A$ be an integral domain (actually $A \neq 0$ suffices!) which is a finitely generated $\mathbb{Z}$-algebra. We want to prove that there is a prime number $p$ such that $A/(p) \neq 0$.

Choose any maximal ideal $\mathfrak{m}$ of $A$. Then $A/\mathfrak{m}$ is a field which is a finitely generated $\mathbb{Z}$-algebra. It is a well-known fact (MO/30599) that this implies that $A/\mathfrak{m}$ is a finite field, say of characteristic $p$. Then $p \in \mathfrak{m}$ and therefore $A/\mathfrak{m}$ surjects onto $A/(p)$, and we are done.