Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, and Anupam Saikia. "Another look at real quadratic fields of relative class number 1." Acta Arithmetica 163.4 (2014): 371-377. <http://eudml.org/doc/286320>.

@article{DebopamChakraborty2014,
abstract = {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.},
author = {Debopam Chakraborty, Anupam Saikia},
journal = {Acta Arithmetica},
keywords = {quadratic fields; fundamental unit; relative class number; quadratic orders; Mersenne primes; Sophie Germain primes},
language = {eng},
number = {4},
pages = {371-377},
title = {Another look at real quadratic fields of relative class number 1},
url = {http://eudml.org/doc/286320},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Debopam Chakraborty
AU - Anupam Saikia
TI - Another look at real quadratic fields of relative class number 1
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 4
SP - 371
EP - 377
AB - 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.
LA - eng
KW - quadratic fields; fundamental unit; relative class number; quadratic orders; Mersenne primes; Sophie Germain primes
UR - http://eudml.org/doc/286320
ER -