Isomorphism of the ideal class group with a cyclic group

Question

Let $K=\mathbb{Q}(\sqrt{-17})$. Show that the ideal class group $Cl_K$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$.

We know that the class number is 4...How i show that $Cl_K$ is cyclic?

Answer 1

We know that $(2)=P^2$ with $P=(2,1+\sqrt{-17})$ since $-17\equiv 3(4)$, and because the Legendre symbol $(-17/3)=1$, we know that $(3)=QQ'$ with $Q=(3,1+\sqrt{-17})$. From $(1+\sqrt{-17})=PQ^2$ and $P^2=(2)\sim (1)$ we obtain $Q^4\sim (1)$. Since there are no elements of order $3$, or $9$ or $27$ it follows that the class of $Q$ generates a cyclic subgroup of order $4$ in $CL(K)$. Since the class number is $4$, it is already the class group.