(a) Let $K$ be a number field of degree $n$. Define the discriminant $\operatorname{disc}\left(\alpha_{1}, \ldots, \alpha_{n}\right)$ of an $n$-tuple of elements $\alpha_{i}$ of $K$, and show that it is nonzero if and only if $\alpha_{1}, \ldots, \alpha_{n}$ is a $\mathbb{Q}$-basis for $K$.

(b) Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial

$$T^{n}+\sum_{j=0}^{n-1} a_{j} T^{j}, \quad a_{j} \in \mathbb{Z}$$

and assume that $p$ is a prime such that, for every $j, a_{j} \equiv 0(\bmod p)$, but $a_{0} \not \equiv 0\left(\bmod p^{2}\right)$.

(i) Show that $P=(p, \alpha)$ is a prime ideal, that $P^{n}=(p)$ and that $\alpha \notin P^{2}$. [Do not assume that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.]

(ii) Show that the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ is prime to $p$.

(iii) If $K=\mathbb{Q}(\alpha)$ with $\alpha^{3}+3 \alpha+3=0$, show that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$. [You may assume without proof that the discriminant of $T^{3}+a T+b$ is $-4 a^{3}-27 b^{2}$.]