Let $\mathcal{O}_{L}$ be the ring of integers in a number field $L$, and let $\mathfrak{a} \leqslant \mathcal{O}_{L}$ be a non-zero ideal of $\mathcal{O}_{L}$.

(a) Show that $\mathfrak{a} \cap \mathbb{Z} \neq\{0\}$.

(b) Show that $\mathcal{O}_{L} / \mathfrak{a}$ is a finite abelian group.

(c) Show that if $x \in L$ has $x \mathfrak{a} \subseteq \mathfrak{a}$, then $x \in \mathcal{O}_{L}$.

(d) Suppose $[L: \mathbb{Q}]=2$, and $\mathfrak{a}=\langle b, \alpha\rangle$, with $b \in \mathbb{Z}$ and $\alpha \in \mathcal{O}_{L}$. Show that $\langle b, \alpha\rangle\langle b, \bar{\alpha}\rangle$ is principal.

[You may assume that $\mathfrak{a}$ has an integral basis.]