A Note on 3-Divisibility of Class Number of Quadratic Field

Abstract

In this paper, the authors show that there exists infinitely many family of pairs of quadratic fields $\mathbb{Q} (\sqrt D)$ and $\mathbb{Q} (\sqrt {D + n})$ with D,n ∈ ℤ whose class numbers are both divisible by 3.

Keywords

Quadratic field / Class number / Hilbert class field

References

[1] Iizuka Y. On the class number divisibility of pairs of imaginary quadratic fields. J. Number Theory, 2018, 184: 122-127

[2] Iizuka Y, Konomi Y, Nakano S. An application of the arithmetic of elliptic curves to the class number problem for quadratic fields. Tokyo J. Math., 2021, 1: 1-15

[3] Kato K, Kurokawa N, Saito T. Number Theory 2: Introduction to Class Field Theory, 2011, Providence: American Mathematical Society

[4] Kishi Y, Miyake K. Parametrization of the quadratic fields whose class numbers are divisible by three. J. Number Theory, 2000, 80: 209-217

[5] Komatsu T. An infinite family of pairs of quadratic fields $\mathbb{Q} (\sqrt D)$ and $\mathbb{Q} (\sqrt {mD})$ whose class numbers are both divisible by 3. Acta Arith., 2002, 104: 129-136

[6] Silverman J. The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, 2009 2nd edition New York: Springer-Verlag

[7] Xie J F, Chao K F. On the divisibility of class numbers of imaginary quadratic fields $(\mathbb{Q} (\sqrt D),\mathbb{Q} (\sqrt {D + m}))$. Ramanujan J., 2020, 53: 517-528