On relative class Number and continued fractions
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1559-1568
https://doi.org/10.4134/BKMS.2015.52.5.1559
Printed September 30, 2015
Debopam Chakraborty and Anupam Saikia
Indian Institute of Technology, Indian Institute of Technology
Abstract : The relative class number $H_{d}(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant $d$ is the ratio of class numbers of $\mathcal{O}_{f}$ and $\mathcal{O}_{K}$, where $\mathcal{O}_{K}$ denotes the ring of integers of $K$ and $\mathcal{O}_{f}$ is the order of conductor $f$ given by $\mathbb{Z}+f\mathcal{O}_{K}$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $\sqrt{m}$ has a diagonal form. Here, we extend their result and show that there exists a conductor $f$ of relative class number $1$ when the continued fraction of $\sqrt{m}$ is non-diagonal of period $4$ or $5$. We also show that there exist infinitely many real quadratic fields with any power of $2$ as relative class number if there are infinitely many Mersenne primes.
Keywords : relative class number, continued fraction
MSC numbers : Primary 11A55, 11R11, 11R65
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