Every prime ideal in $\mathbb{Z}[x]$ is generated by at most two elements [duplicate]
(This is more "things that work in $\mathbb{Z}[x]$" more than "general algebraic methods", but it works.)
Suppose you have three (or more) generators. Then either two of them are constants or two of them are polynomials (of positive degree). With what can you replace these two generators? (Hint: Same answer for both cases.)