Taiwanese Journal of Mathematics

REMARKS ON QUADRATIC FIELDS WITH NONCYCLIC IDEAL CLASS GROUPS

Kwang-Seob Kim

Full-text: Open access

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 $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}  \times \mathbb{Z}/n\mathbb{Z}$.

Article information

Source
Taiwanese J. Math., Volume 19, Number 5 (2015), 1387-1399.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133715

Digital Object Identifier
doi:10.11650/tjm.19.2015.5081

Mathematical Reviews number (MathSciNet)
MR3412012

Zentralblatt MATH identifier
1357.11112

Subjects
Primary: 11R29: Class numbers, class groups, discriminants 11R11: Quadratic extensions

Keywords
ideal class group real quadratic fields

Citation

Kim, Kwang-Seob. REMARKS ON QUADRATIC FIELDS WITH NONCYCLIC IDEAL CLASS GROUPS. Taiwanese J. Math. 19 (2015), no. 5, 1387--1399. doi:10.11650/tjm.19.2015.5081. https://projecteuclid.org/euclid.twjm/1499133715


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References

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