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

## TL;DR

This paper investigates the structure of order 4 class groups in real quadratic fields of the form (+1) and provides conditions to determine their structure, along with computations of Dedekind zeta values and growth of class group size.

## Contribution

It offers new sufficient conditions to determine the structure of class groups of order 4 in a specific family of real quadratic fields and analyzes their size growth.

## Key findings

- Class groups of order 4 are either cyclic or product of two cyclic groups.
- Conditions are provided to specify the structure of these class groups.
- The size of the class group can be increased by adding more odd prime factors to n.

## 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 $n$.