# Determine all the prime ideals of $\Bbb Z[x]$ (maybe a different method) [duplicate]

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?