Principality by reduced ideals in pure cubic number fields

Authors

  • Jamal Benamara Department of Mathematics, Faculty of Sciences, Mohammed First University, 60000 Oujda, Morocco https://orcid.org/0000-0003-0303-0355
  • Mohammed Talbi Regional Center of Education and Training, 60000 Oujda, Morocco

DOI:

https://doi.org/10.33044/revuma.3862

Abstract

This paper describes a method for determining the list of reduced ideals of any pure cubic number field, which we can use for testing the principality of these fields and give a generator for a principal ideal.

Downloads

Download data is not yet available.

References

Ş. Alaca and K. S. Williams, Introductory algebraic number theory, Cambridge University Press, Cambridge, 2004.  MR  Zbl

A. Azizi, J. Benamara, M. C. Ismaili, and M. Talbi, The reduced ideals of a special order in a pure cubic number field, Arch. Math. (Brno) 56 no. 3 (2020), 171–182.  DOI  MR  Zbl

J. Buchmann, On the computation of units and class numbers by a generalization of Lagrange's algorithm, J. Number Theory 26 no. 1 (1987), 8–30.  DOI  MR  Zbl

J. Buchmann and H. C. Williams, A key-exchange system based on imaginary quadratic fields, J. Cryptology 1 no. 2 (1988), 107–118.  DOI  MR  Zbl

H. Cohen, A course in computational algebraic number theory, Grad. Texts in Math. 138, Springer, Berlin, 1993.  DOI  MR  Zbl

B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Transl. Math. Monogr. 10, American Mathematical Society, Providence, RI, 1964.  MR  Zbl

G. T. Jacobs, Reduced ideals and periodic sequences in pure cubic fields, Ph.D. thesis, University of North Texas, 2015.  DOI  MR

R. A. Mollin, Quadratics, Discrete Math. Appl., CRC Press, Boca Raton, FL, 1996.  MR  Zbl

J. Neukirch, Algebraic number theory, Grundlehren Math. Wiss. 322, Springer, Berlin, 1999.  DOI  MR  Zbl

J. J. Payan, Sur le groupe des classes d'un corps quadratique, Cours de l'institut Fourier 7 (1972), 2–30. Available at http://www.numdam.org/item?id=CIF_1972__7__2_0.

C. Prabpayak, Orders in pure cubic number fields, Grazer Math. Ber. 361, Institut für Mathematik, Karl-Franzens-Universität Graz, Graz, 2014.  MR  Zbl

R. Scheidler, J. A. Buchmann, and H. C. Williams, A key-exchange protocol using real quadratic fields, J. Cryptology 7 no. 3 (1994), 171–199.  DOI  MR  Zbl

Downloads

Published

2025-06-02

Issue

Vol. 68 No. 1 (2025)

Section

Article

License

Copyright (c) 2025 Jamal Benamara, Mohammed Talbi

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.