Skip to main content
SpringerLink
Log in

Ideal class groups and their subgroups of real quadratic fields

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z n+t−1)2+4t(witht|z n−1), which contains the well-known fields withm=4z n+1 andm=4z 2n+4 as special cases.

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. Cornell, G, Washington, L., Class numbers of cyclotomic fields,J. Number Theory, 1985, 21:260.

    Article  MATH  MathSciNet  Google Scholar 

  2. Zhang Xianke, Determination of solutions and solvabilities of Diophantine equations and quadratic fields, inAlgebraic Geometry and Algebraic Number Theory (eds. Chern, S. S., Feng Ke-Qin, Li Ke-Zheng), Singapore-London: World Scientific, 1992, 189–199.

    Google Scholar 

  3. Zhang Xianke, Solutions of the Diophantine equations related to real quadratic fields,Chinese Science Bulletin, 1991, 37 (23):1772.

    Google Scholar 

  4. Zhang Xianke, The determinatin of subgroups in ideal class groups of real quadratic fields,Chinese Science Bulletin, 1991, 37 (24):1847.

    Google Scholar 

  5. Zhang Xianke, Criteria of class numbersh (m) = 1 for real quadratic fields,Chinese Science Bulletin, 1992, 38 (1):24.

    Google Scholar 

  6. Katsumi Tanahashi, On certain class numbers of real quadratic fields,Churact. Arith. Fundt., 1976, 274:108.

    MathSciNet  Google Scholar 

  7. Lu Hongwen, Divisibility for class numbers of a kind of real quadratic fields,Acta Math. Sinica, 1985, 28:756.

    MATH  MathSciNet  Google Scholar 

  8. Weinberger, P. J., Real quadratic fields with class numbers divisible by n,J. Number Theory, 1973, 5:237.

    Article  MATH  MathSciNet  Google Scholar 

  9. Louboutin, S., Arithmétique des corps quadratiques réels et fractions continues.Thèse de doctoral, Univ. ParisVII., 1987.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Lawrence C. Washington

  2. Department of Mathematics, Tsinghua University, 100084, Beijing, China

    Xianke Zhang

Authors
  1. Lawrence C. Washington
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianke Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Project supported by the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Washington, L.C., Zhang, X. Ideal class groups and their subgroups of real quadratic fields. Sci. China Ser. A-Math. 40, 909–916 (1997). https://doi.org/10.1007/BF02878670

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02878670

Keywords

Search

Navigation