Another look at real quadratic fields of relative class number 1
Volume 163 / 2014
Abstract
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 $1$.
