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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.

  • 0
    When you say "(1) $P$ contains a prime ideal $p$", what do you mean? This statement has no content-it is always trivially true with $P=p$.2012-07-24
  • 0
    I meant $P$ contains a prime number $p$. I'll edit it later.2012-07-24

2 Answers 2