Indivisibility of Class Numbers of Real Quadratic Fields

Ken Ono

doi:10.1023/A:1001533613223

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the ’Cohen and Lenstra Heuristics‘ give a probability that p[nmid ]h(D). If p>3 is prime, then subject to a mild condition, we show that $\# \{0<D<X|p\nmid h(D)\}\gg_p \frac{\sqrt{X}}{\log X}.$ This condition holds for each 3<p<5000.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Byeon, Dongho 2002. A Note on the Existence of Certain Infinite Families of Imaginary Quadratic Fields. Journal of Number Theory, Vol. 97, Issue. 1, p. 165.

Yu, Gang 2002. A Note on the Divisibility of Class Numbers of Real Quadratic Fields. Journal of Number Theory, Vol. 97, Issue. 1, p. 35.

Byeon, Dongho 2003. Existence of certain fundamental discriminants and class numbers of real quadratic fields. Journal of Number Theory, Vol. 98, Issue. 2, p. 432.

TAYA, Hisao and YAMAMOTO, Gen 2010. On the Density of Real Quadratic Fields with λ2=μ2=ν2=0. Interdisciplinary Information Sciences, Vol. 16, Issue. 1, p. 87.

Fukuda, Takashi Komatsu, Keiichi and Morisawa, Takayuki 2012. On $\lambda$-invariants of $\mathbb{Z}_\ell$-extensions over real abelian number fields of prime power conductors. Functiones et Approximatio Commentarii Mathematici, Vol. 47, Issue. 1,

Ito, Akiko 2013. Existence of an infinite family of pairs of quadratic fields $\mathbb{Q}(\sqrt{m_1D})$ and $\mathbb{Q}(\sqrt{m_2D})$ whose class numbers are both divisible by $3$ or both indivisible by $3$. Functiones et Approximatio Commentarii Mathematici, Vol. 49, Issue. 1,

Lee, Jungyun and Lee, Yoonjin 2016. Indivisibility of class numbers of real quadratic function fields. Journal of Pure and Applied Algebra, Vol. 220, Issue. 8, p. 2828.

Assim, Jilali and Bouazzaoui, Zakariae 2020. Half-integral weight modular forms and real quadratic $p$-rational fields. Functiones et Approximatio Commentarii Mathematici, Vol. 63, Issue. 2,

Benmerieme, Y. and Movahhedi, A. 2021. Multi-quadratic p-rational number fields. Journal of Pure and Applied Algebra, Vol. 225, Issue. 9, p. 106657.

Chattopadhyay, Jaitra and Saikia, Anupam 2022. Simultaneous indivisibility of class numbers of pairs of real quadratic fields. The Ramanujan Journal, Vol. 58, Issue. 3, p. 905.

Chattopadhyay, Jaitra and Saikia, Anupam 2023. On the p-ranks of the ideal class groups of imaginary quadratic fields. The Ramanujan Journal, Vol. 62, Issue. 2, p. 571.

Alkan, Emre 2023. Covering sumsets of a prime field and class numbers. Research in the Mathematical Sciences, Vol. 10, Issue. 4,

Article contents

Indivisibility of Class Numbers of Real Quadratic Fields

Abstract

Keywords

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Indivisibility of Class Numbers of Real Quadratic Fields

Abstract

Keywords

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests