(a) Let $L$ be a number field. State Minkowski's upper bound for the norm of a representative for a given class of the ideal class group $\mathrm{Cl}\left(\mathcal{O}_{L}\right)$.

(b) Now let $K=\mathbb{Q}(\sqrt{-47})$ and $\omega=\frac{1}{2}(1+\sqrt{-47})$. Using Dedekind's criterion, or otherwise, factorise the ideals $(\omega)$ and $(2+\omega)$ as products of non-zero prime ideals of $\mathcal{O}_{K}$.

(c) Show that $\mathrm{Cl}\left(\mathcal{O}_{K}\right)$ is cyclic, and determine its order.

[You may assume that $\left.\mathcal{O}_{K}=\mathbb{Z}[\omega] .\right]$