Advertisement

The Ramanujan Journal

, Volume 46, Issue 3, pp 727–742 | Cite as

Pell-type equations and class number of the maximal real subfield of a cyclotomic field

  • Azizul Hoque
  • Kalyan Chakraborty
Article
  • 363 Downloads

Abstract

We investigate the solvability of the Diophantine equation \(x^2-my^2=\pm p\) in integers for certain integer m and prime p. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly larger than 1.

Keywords

Diophantine equation Real quadratic fields Maximal real subfield of cyclotomic field Class number 

Mathematics Subject Classification

Primary: 11D09 11R29 Secondary: 11R11 11R18 

Notes

Acknowledgements

The authors are indebted to the anonymous referee for his/her valuable suggestions which have helped improving the presentation of this manuscript.

References

  1. 1.
    Ankeny, N.C., Chowla, S., Hasse, H.: On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 217, 217–220 (1965)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dickson, L.E.: History of the Theory of Numbers, vol. 2. Chelsea, New York (1952)Google Scholar
  3. 3.
    Hoque, A., Saikia, H.K.: On the class-number of the maximal real subfield of a cyclotomic field. Quaes. Math. 37(7), 889–894 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kaplan, P., Williams, K.S.: Pell’s equations \(X^2-mY^2=-1, -4\) and continued fractions. J. Number Theory 23, 169–182 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lang, S.D.: Note on the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 290, 70–72 (1977)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Matthews, K.R.: The diophantine equation \(x^2-Dy^2=N, D > 1\), in integers. Expos. Math. 18, 323–331 (2000)zbMATHGoogle Scholar
  7. 7.
    Osada, H.: Note on the class-number of the maximal real subfield of a cyclotomic field. Manuscr. Math. 58, 215–227 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sawilla, R.E., Silvester, A.K., Williams, H.C.: A New Look at an Old Equation. Algorithmic Number Theory (ANTS-VIII). Lecture Notes in Computer Science, vol. 5011. Springer, Berlin, pp. 37–59 (2008)Google Scholar
  9. 9.
    Serret, J.-A. (ed.): Oeuvres de Lagrange. I-XIV. Gauthiers-Villars, Paris (1877)Google Scholar
  10. 10.
    Takeuchi, H.: On the class-number of the maximal real subfield of a cyclotomic field. Can. J. Math. 33(1), 55–58 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yamaguchi, I.: On the class-number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 272, 217–220 (1975)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Harish-Chandra Research Institute, HBNIAllahabadIndia

Personalised recommendations