On the structure of order 4 class groups of $\mathbb{Q}(\sqrt{n^2+1})$

Abstract: Groups of order $4$ are isomorphic to either $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. We give certain sufficient conditions permitting to specify the structure of class groups of order $4$ in the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$ as $n$ varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point $-1$. As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of $n$.