Advertisement

Computing subfields: Reverse of the primitive element problem

  • D. Lazard
  • A. Valibouze
Part of the Progress in Mathematics book series (PM, volume 109)

Abstract

We describe an algorithm which computes all subfields of an effectively given finite algebraic extension. Although the base field can be arbitrary, we focus our attention on the rationals.

This appears to be a fundamental tool for the simplification of algebraic numbers.

Keywords

Symmetric Group Symmetric Function Galois Group Monomial Form Algebraic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Baeckelin, J. and Fröberg, R (1991). How we proved that there are exactly 924 cyclic 7-roots. Proceedings of ISSAC’91 (S.W. Watt ed.). ACM Press (New-York), 103–111.Google Scholar
  2. [2]
    Batu, C., Bernadi, D., Cohen, H. and Olivier, M. (1991). User’s Guide to PARI-GP. Available by anonymous ftp from math.ucla.edu (128.97.4.254).Google Scholar
  3. [3]
    Björck G. and Fröberg, R., (1989), A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots, Reports, Matematiska Institutionen, Stockholms Universitet, 1989-No 7.Google Scholar
  4. [4]
    Davenport, J.H. (1987). Looking at a set of equations. Technical report 87-06, University of Bath.Google Scholar
  5. [5]
    Dixon, J.D. (1990). Computing subfields in algebraic number fields. J. Austral. Math. Soc. (serie A) 49, 434–448.CrossRefzbMATHGoogle Scholar
  6. [6]
    Faugère, J.C., Gianni, P., Lazard, D. and Mora, T. (1989). Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering. Submitted to J. Symb. Comp. Technical Report LITP 89–52.Google Scholar
  7. [7]
    Giusti, M., Lazard, D. and Valibouze, A. (1988). Algebraic transformations of polynomial equations, symmetric polynomials and elimination. Symbolic and Algebraic Computation, International Symposium ISSAC’ 88 (P. Gianni, ed.), Lect. Notes in Comp. Sc. 358, 309–314Google Scholar
  8. [8]
    Kozen, D and Landau, S (1989). Polynomial decomposition algorithms. J. Symb. Comp. 7, 445–456.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Lagrange, J.L., (1770–1771). Réflexions sur la résolution algébrique des équations. Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin.Google Scholar
  10. [10]
    Lazard, D. (1992). Solving zero-dimensional algebraic systems. J. Symbolic Computation 13, 117–131.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Soicher, L. (1981). The computation of the Galois groups.. Thesis, Concordia University, Montreal (Quebec, Canada, 1981).Google Scholar
  12. [12]
    Valibouze, A. (1987). Fonctions symétriques et changements de bases. European Conference on Computer Algebra, Leipzig, GDR, 1987. (J.H. Davenport, ed.) Lect. Notes in Comp. Sc. 378.Google Scholar
  13. [13]
    Valibouze, A. (1989). Symbolic computation with symmetric polynomials, an extension to Macsyma. Computers and Mathematics (1989, MIT, Cambridge, Mass.). Springer-Verlag.Google Scholar
  14. [14]
    Valibouze, A. (1989). Résolvantes et fonctions symétriques. Proc. of the ACM-SIGSAM 1989 Intern. Symp. on Symbolic and Algebraic Computation, ISS AC‘89 (Portland, Oregon). ACM Press, 390–399.Google Scholar
  15. [15]
    Zippel, Book to appear.Google Scholar

Copyright information

© Birkhäuser Boston 1993

Authors and Affiliations

  • D. Lazard
    • 1
  • A. Valibouze
    • 1
  1. 1.LITPUniversité P. et M. CurieParis Cedex 05France

Personalised recommendations