Journal of the Ramanujan Mathematical Society

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A Short Note on the Divisibility of Class Numbers of Real Quadratic Fields


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1 Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
     

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For any integer l ≥ 1, let p1, p2, . . . , pl+2 be distinct prime numbers ≥ 5. For all real numbers X > 1, we let N3,l (X) denote the number of real quadratic fields K whose absolute discriminant dK ≤ X and dK is divisible by (p1 . . . pl+2) together with the class number hK of K divisible by 2l · 3. Then, in this short note, by following the method in [3], we prove that N3,l (X) ≫ X 7/8 for all large enough X’s.
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  • N. Ankeny and S. Chowla, On the divisibility of the class numbers of quadratic fields, Pacific J. Math., 5 (1955) 321–324.

  • D. Byeon, Real quadratic fields with class number divisible by 5 or 7, Manu. Math., 120(2) (2006) 211–215.

  • D. Byeon and E. Koh, Real quadratic fields with class number divisible by 3, Manu. Math., 111 (2003) 261–263.

  • K. Chakraborty and M. Ram Murty, On the number of real quadratic fields with class number divisible by 3, Proc. Amer. Math. Soc., 131 (2002) 41–44.

  • H. Davenort and H. Heilbronn, On the density of discriminants of cubic fields, Proc. Royal Soc. A, 322 (1971) 405–420.

  • T. Honda, On real quadratic fields whose class numbers are multiples of 3, J.Reine Angew. Math, 223 (1968) 101–102.

  • Y. Kishi and K. Miyake, Parametrization of the quadratic fields whose class numbers are divisible by three, J. Number Theory, 80 (2000) 209–217.

  • F. Luca, A note on the divisibility of class numbers of real quadratic fields, C. R. Math. Acad. Sci. Soc. R. Can, 25(3) (2003) 71–75.

  • T. Nagell, Uber die Klassenzahl imaginar quadratischer Zahkorper, Abh. Math. Seminar Univ. Hamburg, 1(1) (1922) 140–150.

  • M. Ram Murty, Exponents of class groups of quadratic fields, Topics in Number theory (University Park, PA, (1997) Math. Appl., Kluwer Acad. Publ., Dordrecht, 467 (1999) 229–239.

  • K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc., 61(2) (2000) 681–690.

  • P. Weinberger, Real quadratic fields with class numbers divisible by n, J. Number Theory, 5 (1973) 237–241.

  • Y. Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka J. Math., 7 (1970) 57–76.

  • G. Yu, A note on the divisibility of class numbers of real quadratic fields, J. Number Theory, 97 (2002) 35–44.


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  • A Short Note on the Divisibility of Class Numbers of Real Quadratic Fields

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Authors

Jaitra Chattopadhyay
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract


For any integer l ≥ 1, let p1, p2, . . . , pl+2 be distinct prime numbers ≥ 5. For all real numbers X > 1, we let N3,l (X) denote the number of real quadratic fields K whose absolute discriminant dK ≤ X and dK is divisible by (p1 . . . pl+2) together with the class number hK of K divisible by 2l · 3. Then, in this short note, by following the method in [3], we prove that N3,l (X) ≫ X 7/8 for all large enough X’s.

References