REMARKS ON QUADRATIC FIELDS WITH NONCYCLIC IDEAL CLASS GROUPS

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 Z/nZ x Z/nZ. 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 Z/nZ x Z/nZ In addition, we will prove that there exist infinitely many imaginary quadratic number fields with the ideal class group having a subgroup isomorphic to Z/nZ x Z/nZ x Z/nZ.