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Another look at real quadratic fields of relative class number 1

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Chakraborty D., Saikia A.

Acta Arithmetica

2014

tom 163

nr 4

371-377

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Chakraborty D.

1 Saikia A.

1 2014

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Another look at real quadratic fields of relative class number 1