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I know that there are quite many questions about the prime ideals of $\Bbb Z[x]$ on MSE. Actually I meet an exercise on this (in the book "Algebra: Fields and Galois Theory" by Falko Lorenz, Exercise 5.12) and the author outline an approach as follows:
(a) For every prime number $p$ and every normalized (that is, "monic") $f$ $\in$ $\Bbb Z[x]$ whose image $\bar f$ is irreducible in $\Bbb F_p[x]$, the ideal ($p$, $f$) is a maximal ideal of $\Bbb Z[x]$.
(b) If $P$ is a prime ideal of $\Bbb Z[x]$ that is not of the form given in (a), $P$ is a principal ideal of $\Bbb Z[x]$. If $P$ is a principal ideal of $\Bbb Z[x]$, then $P$ is not a maximal ideal.
I have done by myself (a). For (b), I reduce it into two cases depending on whether $P$ is principal. I also show that if $P$ is principal, then $P$ is not maximal. So where I get confused is the case when $P$ is not principal. I can't reach a contradiction if I further suppose $P$ is prime but not the one in (a).
Any ideas or hints?