How can you show that the non-principal prime ideals of $\mathbb{Z}[x]$ can be generated by only two elements, a prime number $p$ and an irreducible polynomial not in $p\mathbb{Z}[x]$?
I can get to the point in the proof that a prime ideal with more than one generator must contain some $p$, but I can't prove that appending the polynomial can generate the prime ideal itself.
HINT $\ $ The image of the ideal in $\rm\:\mathbb F_p[x]$ is principal. Pull this information back to $\rm\:\mathbb Z[x]\:.$