Skip to main content
SpringerLink
Log in

Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

In this article, we present new results for efficient arithmetic operations in a number field K represented by successive extensions. These results are based on multi-modular and evaluation–interpolation techniques. We show how to use intrinsic symmetries in order to increase the efficiency of these techniques. Applications to splitting fields of univariate polynomials are presented.

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. Abdeljaouad-Tej I., Orange S., Renault G., Valibouze A.: Computation of the decomposition group of a triangular ideal. Appl. Algebra Eng. Commun. Comput. 15(3–4), 279–294 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anai, H., Noro, M., Yokoyama, K.: Computation of the splitting fields and the Galois groups of polynomials. In: Algorithms in Algebraic Geometry and Applications (Santander, 1994). Progr. Math., vol. 143, pp. 29–50. Birkhäuser, Basel (1996)

  3. Aubry A., Valibouze A.: Using Galois ideals for computing relative resolvents. J. Symb. Comput. 30(6), 635–651 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997) [Computational algebra and number theory (London, 1993)]

    Article  MathSciNet  MATH  Google Scholar 

  5. Bostan A., Chowdhury M., van der Hoeven J., Schost E.: Homotopy techniques for multiplication modulo triangular sets. J. Symb. Comput. 46(12), 1378–1402 (2011)

    Article  MATH  Google Scholar 

  6. Cohen, H.: A course in computational algebraic number theory. In: Graduate Texts in Mathematics, vol. 138. Springer-Verlag, Berlin (1993)

  7. Cox, J., Little, D., O’Shea, D.: Ideals, varieties, and algorithms. In: Undergraduate Texts in Mathematics Springer-Verlag, NewYork (1991)

  8. Dahan, X., Schost, É.: Sharp estimates for triangular sets. In: ISSAC’04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, pp. 103–110. ACM (2004)

  9. Diaz-Toca G.M., Lombardi H.: Dynamic Galois Theory. J. Symb. Comput. 45(12), 1316–1329 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ducos L.: Construction de corps de décomposition grâce aux facteurs de résolvantes. Commun. Algebra 28(2), 903–924 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Geissler K., Klüners J.: Galois group computation for rational polynomials. J. Symb. Comput. 30(6), 653–674 (2000) (Algorithmic methods in Galois theory)

    Article  MATH  Google Scholar 

  12. Langemyr, L.: Algorithms for a multiple algebraic extension. In: Effective Methods in Algebraic Geometry. Progress in Mathematics, vol. 94, pp. 235–248. Birkhäuser, Boston (1991)

  13. Langemyr, L.: Algorithms for a multiple algebraic extension II. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 539, pp. 234–233. Springer-Verlag, Berlin (1991)

  14. Lederer M.: Explicit constructions in splitting fields of polynomials. Riv. Mat. Univ. Parma (7) 3*, 233–244 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Li X., Moreno Maza M., Schost É.: Fast arithmetic for triangular sets: from theory to practice. J. Symb. Comput. 44(7), 891–907 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Orange, S.: Calcul de corps de décomposition—Utilisations fines d’ensembles de permutations en théorie de Galois effective. Ph.D. thesis, LIP6, Université Paris VI (2006)

  17. Orange, S., Renault, G., Yokoyama, K.: Computation schemes for splitting fields of polynomials. In: ISSAC ’09: Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pp. 279–286. ACM (2009)

  18. Pohst, M., Zassenhaus, H.: Algorithmic algebraic number theory. In: Encyclopedia of Mathematics and its Applications, vol. 30. Cambridge University Press, Cambridge (1997) (Revised reprint of the 1989 original)

  19. Renault, G., Yokoyama, K.: A modular method for computing the splitting field of a polynomial. In: Algorithmic Number Theory Symposium, vol. 4076 (2006)

  20. Renault, G., Yokoyama, K.: Multi-modular algorithm for computing the splitting field of a polynomial. In: ISSAC ’08: Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, pp. 247–254. ACM (2008)

  21. Rennert N., Valibouze A.: Calcul de résolvantes avec les modules de Cauchy. Exp. Math. 8(4), 351–366 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yokoyama K.: A modular method for computing the Galois groups of polynomials. J. Pure Appl. Algebra 117/118, 617–636 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMAH, Université du Havre, 25 rue Philippe Lebon, 76600, Le Havre, France

    Sébastien Orange

  2. INRIA, Paris-Rocquencourt, POLSYS Project, CNRS, UMR 7606, LIP6, UPMC, Univ. Paris 06, Case 169, 4, Place Jussieu, 75252, Paris, France

    Guénaël Renault

  3. Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Kazuhiro Yokoyama

Authors
  1. Sébastien Orange
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guénaël Renault
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kazuhiro Yokoyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Sébastien Orange, Guénaël Renault or Kazuhiro Yokoyama.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Orange, S., Renault, G. & Yokoyama, K. Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries. Math.Comput.Sci. 6, 217–233 (2012). https://doi.org/10.1007/s11786-012-0112-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-012-0112-y

Keywords

Mathematics Subject Classification

Search

Navigation