Let $K$ be a finite extension of $\mathbb{Q}$ and let $\mathcal{O}=\mathcal{O}_{K}$ be its ring of integers. We will assume that $\mathcal{O}=\mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}$. The minimal polynomial of $\theta$ will be denoted by $g$. For a prime number $p$ let

$$\bar{g}(X)=\bar{g}_{1}(X)^{e_{1}} \cdot \ldots \cdot \bar{g}_{r}(X)^{e_{r}}$$

be the decomposition of $\bar{g}(X)=g(X)+p \mathbb{Z}[X] \in(\mathbb{Z} / p \mathbb{Z})[X]$ into distinct irreducible monic factors $\bar{g}_{i}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$. Let $g_{i}(X) \in \mathbb{Z}[X]$ be a polynomial whose reduction modulo $p$ is $\bar{g}_{i}(X)$. Show that

$$\mathfrak{p}_{i}=\left[p, g_{i}(\theta)\right], \quad i=1, \ldots, r,$$

are the prime ideals of $\mathcal{O}$ containing $p$, that these are pairwise different, and

$$[p]=\mathfrak{p}_{1}^{e_{1}} \cdot \ldots \cdot \mathfrak{p}_{r}^{e_{r}}$$