Suppose that $K$ is a number field with ring of integers $\mathcal{O}_{K}$.

(i) Suppose that $M$ is a sub- $\mathbb{Z}$-module of $\mathcal{O}_{K}$ of finite index $r$ and that $M$ is closed under multiplication. Define the discriminant of $M$ and of $\mathcal{O}_{K}$, and show that $\operatorname{disc}(M)=r^{2} \operatorname{disc}\left(\mathcal{O}_{K}\right) .$

(ii) Describe $\mathcal{O}_{K}$ when $K=\mathbb{Q}[X] /\left(X^{3}+2 X+1\right)$.

[You may assume that the polynomial $X^{3}+p X+q$ has discriminant $-4 p^{3}-27 q^{2}$.]

(iii) Suppose that $f, g \in \mathbb{Z}[X]$ are monic quadratic polynomials with equal discriminant $d$, and that $d \notin\{0,1\}$ is square-free. Show that $\mathbb{Z}[X] /(f)$ is isomorphic to $\mathbb{Z}[X] /(g)$.

[Hint: Classify quadratic fields in terms of discriminants.]