State Minkowski's theorem.

Let $K=\mathbb{Q}(\sqrt{-d})$, where $d$ is a square-free positive integer, not congruent to 3 $(\bmod 4) .$ Show that every nonzero ideal $I \subset \mathcal{O}_{K}$ contains an element $\alpha$ with

$$0<\left|N_{K / \mathbb{Q}}(\alpha)\right| \leqslant \frac{4 \sqrt{d}}{\pi} N(I)$$

Deduce the finiteness of the class group of $K$.

Compute the class group of $\mathbb{Q}(\sqrt{-22})$. Hence show that the equation $y^{3}=x^{2}+22$ has no integer solutions.