Suppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

(i) Taking for granted the fact that there is a constant $C_{K}$ such that every integral ideal $I$ of $\mathcal{O}_{K}$ has a non-zero element $x$ such that $|N(x)| \leqslant C_{K} N(I)$, deduce that the class group of $K$ is finite.

(ii) Compute the class group of $\mathbb{Q}(\sqrt{-21})$, given that you can take

$$C_{K}=\left(\frac{4}{\pi}\right)^{s} \frac{n !}{n^{n}}\left|D_{K}\right|^{1 / 2}$$

where $D_{K}$ is the discriminant of $K$.

(iii) Find all integer solutions of $y^{2}=x^{3}-21$.