By Dedekind's theorem, or otherwise, factorise $2,3,5$ and 7 into prime ideals in the field $K=\mathbb{Q}(\sqrt{-34})$. Show that the ideal equations

$$[\omega]=[5, \omega][7, \omega], \quad[\omega+3]=[2, \omega+3][5, \omega+3]^{2}$$

hold in $K$, where $\omega=1+\sqrt{-34}$. Hence, prove that the ideal class group of $K$ is cyclic of order $4 .$

[It may be assumed that the Minkowski constant for $K$ is $2 / \pi$.]