(a) Let $L$ be a number field, and suppose there exists $\alpha \in \mathcal{O}_{L}$ such that $\mathcal{O}_{L}=\mathbb{Z}[\alpha]$. Let $f(X) \in \mathbb{Z}[X]$ denote the minimal polynomial of $\alpha$, and let $p$ be a prime. Let $\bar{f}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$ denote the reduction modulo $p$ of $f(X)$, and let

$$\bar{f}(X)=\bar{g}_{1}(X)^{e_{1}} \cdots \bar{g}_{r}(X)^{e_{r}}$$

denote the factorisation of $\bar{f}(X)$ in $(\mathbb{Z} / p \mathbb{Z})[X]$ as a product of powers of distinct monic irreducible polynomials $\bar{g}_{1}(X), \ldots, \bar{g}_{r}(X)$, where $e_{1}, \ldots, e_{r}$ are all positive integers.

For each $i=1, \ldots, r$, let $g_{i}(X) \in \mathbb{Z}[X]$ be any polynomial with reduction modulo $p$ equal to $\bar{g}_{i}(X)$, and let $P_{i}=\left(p, g_{i}(\alpha)\right) \subset \mathcal{O}_{L}$. Show that $P_{1}, \ldots, P_{r}$ are distinct, non-zero prime ideals of $\mathcal{O}_{L}$, and that there is a factorisation

$$p \mathcal{O}_{L}=P_{1}^{e_{1}} \cdots P_{r}^{e_{r}}$$

and that $N\left(P_{i}\right)=p^{\operatorname{deg} \bar{g}_{i}(X)}$.

(b) Let $K$ be a number field of degree $n=[K: \mathbb{Q}]$, and let $p$ be a prime. Suppose that there is a factorisation

$$p \mathcal{O}_{K}=Q_{1} \cdots Q_{s}$$

where $Q_{1}, \ldots, Q_{s}$ are distinct, non-zero prime ideals of $\mathcal{O}_{K}$ with $N\left(Q_{i}\right)=p$ for each $i=$ $1, \ldots, s$. Use the result of part (a) to show that if $n>p$ then there is no $\alpha \in \mathcal{O}_{K}$ such that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.