Skip to main content
SpringerLink
Log in

The narrow class groups of the ℤ17- and ℤ19-extensions over the rational field

  • Published:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Aims and scope Submit manuscript

Abstract

Let p be either 17 or 19, let ℤ p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p 2. We shall prove, with the help of a computer, that the l-class group of the ℤ p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ p -extension.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armitage, J.V., Fröhlich, A.: Classnumbers and unit signatures. Mathematika 14, 94–98 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Hasse, H.: Über die Klassenzahl abelscher Zahlkörper. Akademie-Verlag, Berlin (1952). Springer, Berlin (1985)

    MATH  Google Scholar 

  3. Horie, K.: Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field. J. Lond. Math. Soc. 66, 257–275 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Horie, K.: Primary components of the ideal class groups of the Z p -extension over Q for typical inert primes. Proc. Jpn. Acad. Ser. A 81, 40–43 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Horie, K.: Certain primary components of the ideal class group of the Z p -extension over the rationals. Tohoku Math. J. 59, 259–291 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Horie, K., Horie, M.: The narrow class groups of some ℤ p -extensions over the rationals. Acta Arith. 135, 159–180 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Iwasawa, K.: A note on class numbers of algebraic number fields. Abh. Math. Semin. Univ. Hambg. 20, 257–258 (1956)

    MATH  MathSciNet  Google Scholar 

  8. Washington, L.C.: Class numbers and ℤ p -extensions. Math. Ann. 214, 177–193 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  9. Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. GTM, vol. 83. Springer, New York (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tokai University, 1117 Kitakaname, Hiratsuka, 259-1292, Japan

    Kuniaki Horie

  2. Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan

    Mitsuko Horie

Authors
  1. Kuniaki Horie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mitsuko Horie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mitsuko Horie.

Additional information

Communicated by U. Kühn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horie, K., Horie, M. The narrow class groups of the ℤ17- and ℤ19-extensions over the rational field. Abh. Math. Semin. Univ. Hambg. 80, 47–57 (2010). https://doi.org/10.1007/s12188-009-0030-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12188-009-0030-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation