REMARKS ON QUADRATIC FIELDS WITH NONCYCLIC IDEAL CLASS GROUPS

Abstract

Let $n$ be an integer. Then, it is well known that there are infinitely many imaginary quadratic fields with an ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$. Less is known for real quadratic fields, other than the cases that $n=3,5,$ or $7$, due to Craig [3] and Mestre [4, 5]. In this article, we will prove that there exist infinitely many real quadratic number fields with the ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ In addition, we will prove that there exist infinitely many imaginary quadratic number fields with the ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$.