Skip to main content
SpringerLink
Log in

A rational reciprocity law over function fields

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

In the classical case, reciprocity laws for power residue symbols are called rational, which means that the power residue symbols only assume the values \({\pm 1}\) and have entries in \({\mathbb{Z}}\). We establish a rational reciprocity law over function fields.

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. Artin E.: Quadratische Körper im Gebiete der höheren Kongruenzen. Math. Z. 19, 153–246 (1924)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown E.: Quadratic forms and biquadratic reciprocity. J. Reine Angew. Math. 253, 214–220 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Burde K.: Ein rationales biquadratisches Reziprozitätsgesetz. J. Reine Angew. Math. 235, 175–184 (1969)

    MathSciNet  MATH  Google Scholar 

  4. Dedekind R.: Abriß einer Theorie der höheren Congruenzen in Bezug auf einen reellen Primzahl-Modulus. J. Reine Angew. Math. 54, 1–26 (1857)

    Article  MathSciNet  Google Scholar 

  5. Helou C.: On rational reciprocity. Proc. Amer. Math. Soc. 108, 861–866 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hsu C.-H.: On polynomial reciprocity law. J. Number Theory 101, 13–31 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lehmer E.: On the quadratic character of some quadratic surds. J. Reine Angew. Math. 250, 42–48 (1971)

    MathSciNet  MATH  Google Scholar 

  8. Lehmer E.: Rational reciprocity laws. Amer. Math. Monthly 85, 467–472 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lemmermeyer F.: Rational quartic reciprocity. Acta Arithmetica 67, 387–390 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Lemmermeyer F.: Reciprocity Laws From Euler to Einstein. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  11. M. Rosen, Number Theory in Function Fields, Springer, New York, 2002.

  12. F.K. Schmidt, Zur Zahlentheorie in Körpern von der Charakteristik p. Sitzungsberichte Erlangen 58–59 (1928), 159–172

  13. Williams K.: A rational octic reciprocity law. Pacific J. Math. 63, 563–570 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Williams K., Hardy K., Friesen C.: On the evaluation of the Legendre symbol \({\left(\frac{A+B\sqrt{m}}{p}\right)}\). Acta Arithmetica 45, 255–272 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, Ridai-cho 1-1, Okayama, 700–0005, Japan

    Yoshinori Hamahata

Authors
  1. Yoshinori Hamahata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshinori Hamahata.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamahata, Y. A rational reciprocity law over function fields. Arch. Math. 108, 233–240 (2017). https://doi.org/10.1007/s00013-016-0990-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-016-0990-3

Mathematics Subject Classification

Keywords

Search

Navigation