Another look at real quadratic fields of relative class number 1

Debopam Chakraborty¹ ; Anupam Saikia¹

¹ Department of Mathematics Indian Institute of Technology, Guwahati Guwahati 781039, Assam, India

Acta Arithmetica, Tome 163 (2014) no. 4, pp. 371-377

Résumé

The relative class number $H_{d}(f)$ of a real quadratic field $K=\mathbb {Q}(\sqrt {m})$ of discriminant $d$ is defined to be the ratio of the 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 $\mathbb {Z}+f\mathcal {O}_{K}$. R. Mollin has shown recently that almost all real quadratic fields have relative class number $1$ for some conductor. In this paper we give a characterization of real quadratic fields with relative class number $1$ through an elementary approach considering the cases when the fundamental unit has norm $1$ and norm $-1$ separately. When $\xi _{m}$ has norm $-1$, we further show that if $d$ is a quadratic non-residue modulo a Mersenne prime $f$ then the conductor $f$ has relative class number $1$. We also prove that if $\xi _{m}$ has norm $-1$ and $f$ is a sufficiently large Sophie Germain prime of the first kind such that $d$ is a quadratic residue modulo $2f+1$, then the conductor $2f+1$ has relative class number $1$.

Détail

DOI : 10.4064/aa163-4-5

Keywords: relative class number real quadratic field mathbb sqrt discriminant defined ratio class numbers mathcal mathcal where mathcal denotes ring integers mathcal order conductor given mathbb mathcal nbsp mollin has shown recently almost real quadratic fields have relative class number conductor paper characterization real quadratic fields relative class number through elementary approach considering cases fundamental unit has norm norm separately has norm further quadratic non residue modulo mersenne prime conductor has relative class number nbsp prove has norm sufficiently large sophie germain prime first kind quadratic residue modulo conductor has relative class number nbsp