The relative class number $H_d\left(f\right)$ 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 $\mathbb{Z}+f_K$. 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...

Fix an element $\alpha$ in a quadratic field $K$. Define $S$ as the set of rational primes $p$, for which $\alpha$ has maximal order modulo $p$. Under the assumption of the generalized Riemann hypothesis, we show that $S$ has a density. Moreover, we give necessary and sufficient conditions for the density of $S$ to be positive.