Divisibility of the class numbers of imaginary quadratic fields

TL;DR

This paper constructs families of imaginary quadratic fields with class groups containing a subgroup isomorphic to a given cyclic group of odd order, expanding understanding of class number divisibility.

Contribution

It introduces explicit families of imaginary quadratic fields with class groups divisible by a specified odd integer n, using forms like .

Findings

Constructed families of fields with class number divisible by n

Identified subgroups isomorphic to  in class groups

Extended knowledge of class number divisibility properties

Abstract

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.