Another look at real quadratic fields of relative class number 1

• Autorzy: Debopam Chakraborty , Anupam Saikia
• Języki publikacji: EN

Abstrakty

EN

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 $𝓞_{f}$ and $𝓞_{K}$, where $𝓞_{K}$ denotes the ring of integers of K and $𝓞_{f}$ is the order of conductor f given by $ℤ + f𝓞_{K}$. 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.

Kategorie tematyczne

• 11R11: Quadratic extensions
• 11R65: Class groups and Picard groups of orders

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 4
• Strony: 371-377

Daty

wydano

2014

Twórcy

autor

Debopam Chakraborty

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

autor

Anupam Saikia

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

Identyfikatory

DOI

10.4064/aa163-4-5