Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $A = \mathbb{Z}[X]/(f(X))$. Let $\theta$ = $X$ (mod $f(X)$).
My question: Is the following proposition correct? If yes, how would you prove this?
Proposition Let $P$ be a non-zero prime ideal of $A$. Then the following assertions hold.
(1) $P$ contains a prime number $p$.
(2) One of the following two cases occurs.
a. If $f(X)$ is irreducible mod $p$, then $P = (p)$.
b. If $f(X)$ is not irreducible mod $p$, then $P = (p, g(\theta))$, where $g(X)$ is an irreducible factor of $f(X)$ mod $p$.
(3) $P$ is a maximal ideal and $A/P$ is a finite field of characteristic $p$.
This is a related question.