Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 1-14, 2023

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra

Received July 28, 2021. Published online December 7, 2022.

Abstract: Let $d$ be a square-free positive integer and $h(d)$ be the class number of the real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g \mid h(d)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb{Q}(\sqrt{n^{2g}+4})$. As another application, we deduce some criteria for a class group of prime power order to be cyclic.

Keywords: real quadratic field; class group; class number; Dedekind zeta values

Classification MSC: 11R29, 11R42, 11R11

DOI: 10.21136/CMJ.2022.0264-21

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Mohit Mishra, Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India, current affiliation: Department of Mathematics, Indian Institute of Technology Kanpur, Kalyanpur, Kanpur, Uttar Pradesh 208016, India, e-mail: m.mishra0808@gmail.com

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