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ON THE IDEAL CLASS GROUP OF CERTAIN QUADRATIC FIELDS

  • YASUHIRO KISHI (a1)

Abstract

Let n(≥ 3) be an odd integer. Let k:= be the imaginary quadratic field and k′:= the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

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References

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1.Erickson, C., Kaplan, N., Mendoza, N., Pacelli, A. M and Shayler, T., Parameterized families of quadratic number fields with 3-rank at least 2, Acta Arith. 130 (2007), 141147.
2.Hasse, H., Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Z. 31 (1930), 565582.
3.Kishi, Y., A criterion for a certain type of imaginary quadratic fields to have 3-ranks of the ideal class groups greater than one, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 9397.
4.Kishi, Y., A constructive approach to Spiegelung relations between 3-ranks of absolute ideal class groups and congruent ones modulo (3)2 in quadratic fields, J. Number Theory 83 (2000), 149.
5.Kishi, Y., Note on the divisibility of the class number of certain imaginary quadratic fields, Glasgow Math. J. 51 (2009), 187191.
6.Kishi, Y. and Miyake, K., Parametrization of the quadratic fields whose class numbers are divisible by three, J. Number Theory 80 (2000), 209217.
7.Llorente, P. and Nart, E., Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc. 87 (1983), 579585.
8.Scholz, A., Über die Beziehung der Klassenzahl quadratischer Körper zueinander, J. Reine Angew. Math. 166 (1932), 201203.
9.Yamamoto, Y., On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 5776.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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