Formal computation of Galois groups with relative resolvents

  • Antoine Colin
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


We propound a systematic and formal method to compute the Galois group of a non-necessarily irreducible polynomial: we proceed by successive inclusions, using mostly computations on scalars (and very few on polynomials). It is based on a formal method of specialization of relative resolvents: it consists in expressing the generic coefficients of the resolvent using the powers of a primitive element, thanks to a quadratic space structure; this reduces the problem to that of specializing a primitive element, which we are able to do in the case of the descending by successive inclusions. We incidentally supply a way to make separable a resolvent.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anai, H., Noro, M., Yokoyama, K.: Computation of the Splitting Fields and the Galois Groups of Polynomials. MEGA'94.Google Scholar
  2. 2.
    Arnaudiès, J.M., Valibouze, A.: Résolvantes de Lagrange. Rapport interne LITP 93.63, December 1993.Google Scholar
  3. 3.
    AXIOM, The Scientific Computation System. Jenks, R.D., Sutor, R.S. Springer-Verlag, 1992.Google Scholar
  4. 4.
    Berwick, E.H.: The Condition That A Quintic Equation Should Be Soluble By Radicals. Proc. London Math. Soc. (2) 14 (1995) 301–307.Google Scholar
  5. 5.
    Berwick, E.H.: On Soluble Sextic Equations. Proc. London Math. Soc. (2) 29 (1929), 1–28.Google Scholar
  6. 6.
    Bourbaki, N.: Eléments de mathématiques. Masson, Paris, 1981.Google Scholar
  7. 7.
    Eichenlaub, Y., Olivier, M.: Computation of Galois Groups for Polynomials with Degree up to Eleven. preprint received in June 1994.Google Scholar
  8. 8.
    G.A.P. Groups, Algorithms and Programming. Martin Schönert and others, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, 1992. Scholar
  9. 9.
    Girstmair, K.: On Invariant Polynomials and their Application in Field Theory. Maths of Comp., vol. 48, no 178 (1987), 781–797.Google Scholar
  10. 10.
    de Lagrange, J.L.: Réflexions sur la résolution algébrique des équations. Prussian Academy, 1770.Google Scholar
  11. 11.
    de Lagrange, J.L.: Réflexions sur la résolution algébrique des équations. Mémoires de l'Académie de Berlin, 205–421, Oeuvres de Lagrange, tome IV, 205–421.Google Scholar
  12. 12.
    de Lagrange, J.L.: Traité de la résolution des équations numériques: Notes sur la théorie des équations algébriques. Oeuvres de Lagrange, Tome VIII, 133–367.Google Scholar
  13. 13.
    McKay, J., Soicher, L.: Computing Galois Groups over the Rationals. Journal of number theory 20 (1985) 273–281.Google Scholar
  14. 14.
    Soicher, L.: The Computation of Galois Groups. Thesis, Concordia University (Montreal), April 1981.Google Scholar
  15. 15.
    Stauduhar, R.P.: The Computation of Galois Groups. Math. Comp. 27 (1973) 981–996.Google Scholar
  16. 16.
    Valibouze, A.: Computation of the Galois Groups of the Resolvent Factors for the Direct and Inverse Galois Problems. In this volume, AAECC 1995.Google Scholar
  17. 17.
    Valibouze, A.: Mémoire d'habilitation, December 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Antoine Colin
    • 1
  1. 1.Centre de mathématiques (CNRS URA 169) Ecole PolytechniqueGAGEPalaiseau CedexFrance

Personalised recommendations