Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Question

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks

Answer 1

For real quadratic fields the Minkowski bound is $\frac{1}{2}\sqrt{d_k}$, where $d_k$ is the discriminant. For $K=\mathbb{Q}(\sqrt{10})$ we have $d_k=40$, hence the Minkowski bound is about $3.16227766$. Hence every ideal class contains an integral ideal $I$ with norm $N(I)\le 3$. The class group is generated by prime ideals of norm less than or equal to $3$. Now continue in the same way as here, or here to see that the class group is cyclic of order $2$: the relevant factorizations are, with $\alpha=\sqrt{10}$, $$ (2)=(2,\alpha)^2,\quad (3)=(3,\alpha+1)(3,\alpha+2) $$ Both are not principal ideals, and the ideals on the right hand side are all equivalent. Now $P=(2,\alpha)$ has order $2$ and generates the class group.