Another look at real quadratic fields of relative class number 1

Debopam Chakraborty ; Anupam Saikia

Acta Arithmetica, Tome 166 (2014), p. 371-377 / Harvested from The Polish Digital Mathematics Library

Résumé

The relative class number $H_{d}(f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$, where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$. 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 ξₘ 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 ξₘ 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

Publié le : 2014-01-01

Zbl 1300.11105

EUDML-ID : urn:eudml:doc:286320