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Another look at real quadratic fields of relative class number 1

Tom 163 / 2014

Debopam Chakraborty, Anupam Saikia Acta Arithmetica 163 (2014), 371-377 MSC: Primary 11R11; Secondary 11R65. DOI: 10.4064/aa163-4-5

Streszczenie

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 $1$.

Autorzy

  • Debopam ChakrabortyDepartment of Mathematics
    Indian Institute of Technology, Guwahati
    Guwahati 781039, Assam, India
    e-mail
  • Anupam SaikiaDepartment of Mathematics
    Indian Institute of Technology, Guwahati
    Guwahati 781039, Assam, India
    e-mail

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