My attempt: if $p$ is prime, then $\mathbb{Z}_p$ is a field and the nonzero prime ideals have the form $(f)$, $f$ irreducible in $\mathbb{Z}_p[x]$ (is there a way to find them all?); else we can write $p=p_1^{e_1}\cdots p_n^{e_n}$ and by the Chinese Remainder Theorem we have $\mathbb{Z}/(p)\simeq \mathbb{Z}/(p_1^{e_1}) \times \dots \times \mathbb{Z}/(p_n^{e_n})$. I was planning to use it somehow...
Is it true that $\mathbb{Z}/(p)[x]\simeq \mathbb{Z}/(p_1^{e_1})[x] \times \dots \times \mathbb{Z}/(p_n^{e_n})[x]$?
If afirmative and I could find the prime ideals of each $\mathbb{Z}/(p_i^{e_i})[x]$ then the prime ideals of $\mathbb{Z}/(p)[x]$ would be $\mathbb{Z}/(p_1^{e_1})[x] \times \dots \times \mathrm{Spec}(\mathbb{Z}/(p_i^{e_i})[x])\times \dots \times \mathbb{Z}/(p_n^{e_n})[x]$. Among these, I really don't know which are maximal.
What I wrote is right?
How can I continue?
How can I find the maximals?
Thanks!