Skip to main content
SpringerLink
Log in

Techniques for the Computation of Galois Groups

  • Conference paper
Algorithmic Algebra and Number Theory

Abstract

Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anai, H., Noro, M., Yokoyama, K.: Computation of the splitting fields and Galois groups of polynomials. InAlgorithms in Algebraic Geometry and Applications (L. Gonzales-Vega, T. Recio, eds.), volume 145 of Progress in Mathematics. Birkhäuser, Boston, 1996 22–50

    Google Scholar 

  2. Arnaudies, J.-M., Valibouze, A.: Groupes de Galois de polynomes en degre 10 ou 11. Rapport interne 94.50, Laboratoire informatique theoretique et programmation, Universite Paris VI, (1994)

    Google Scholar 

  3. Arnaudies, J.-M., Valibouze, A.: Lagrange resolvents. In Cohen and Roy [8], 1997 23–40

    MathSciNet  Google Scholar 

  4. Batut, C., Bernardi, D., Cohen, H., Olivier, M.: User’s guide to pari-gpo Technical report, Universite Bordeaux I, (1993)

    Google Scholar 

  5. Casperson, D., Ford, D., McKay, J.: An ideal decomposition algorithm. J. Symb. Comput. 21 (1996) 133–137

    Article  MathSciNet  MATH  Google Scholar 

  6. Casperson, D., McKay, J.: Symmetric functions, m-Sets and Galois groups. Math. Compo 63 (1994) 208 749–757

    MathSciNet  MATH  Google Scholar 

  7. Char, B.W., Geddes, K. 0., Gonnet, G.H., Monagan, M.B., Watt, S. M.: MAPLE - Reference Manual, 5th edition. University of Waterloo, (1988)

    Google Scholar 

  8. Cohen, A.M., Roy, M.-F. (eds.):Proceedings MEGA’96, volume 117-118 of J. Pure Appl. Algebra, (1997)

    Google Scholar 

  9. Colin, A.: Formal computation of Galois groups with relative resolvents. In Applied algebra, Algebraic algorithms and Error-correcting Codes (G. Cohen, M. Giustini, T. Mora, eds.), volume 948 of Lecture Notes in Computer Science. Springer, Heidelberg, 1995 169–182

    Google Scholar 

  10. Colin, A.: Relative resolvents and partition tables in Galois group computations. In Küchlin [30J, 1997 169–182 76 A. Hulpke

    Google Scholar 

  11. Colin, A.: Theorie des invariants effective. Applications it la theorie de Galois et it la resolution de systemes algebriques. Implantation en AXIOM. Ph.D. thesis, Ecole poly technique, (1997)

    Google Scholar 

  12. Conway, J. H., Hulpke, A., McKay, J.: On transitive permutation groups. to appear in LMS Journal of Computation and Mathematics 0

    Google Scholar 

  13. Daberkow, M., Fieker, C., Kliiners, J., Pohst, M., Roegner, K., Schornig, M., Wildanger, K.: KANT V4. J. Symb. Comput. 24 (1997) 267–283

    Google Scholar 

  14. Darmon, H., Ford, D.: Computational verification of Mll and M12 as Galois groups over Q. Comm. Algebra 17 (1989) 2941–2943

    Article  MathSciNet  MATH  Google Scholar 

  15. Davenport, J. H., Smith, G.: Fast recognition of symmetric and alternating Galois groups, (to appear)

    Google Scholar 

  16. Dixon, J. D.: Computing subfields in algebraic number fields. J. Austral. Math. Soc. Ser. A 49 (1990) 434–448

    Article  MATH  Google Scholar 

  17. Eichenlaub, Y.: Problemes effectifs de theorie de Galois en degres 8 it 11. Ph.D. thesis, Universite Bordeaux I, (1996)

    Google Scholar 

  18. Eichenlaub, Y., Olivier, M.: Computation of Galois groups for polynomials with degree up to eleven. Preprint, Universite Bordeaux I, (1995)

    Google Scholar 

  19. Geissler, K.: Zur Berechnung von Galoisgruppen. Diplomarbeit, Fachbereich Mathematik der TU Berlin, (1997)

    Google Scholar 

  20. Geyer, H.: Programme zur Berechnung der Galoisgruppen von Polynomen 8. und 9. Grades. Preprint 93–10, IWR Heidelberg, (1993)

    Google Scholar 

  21. Girstmair, K.: On the computation of resolvents and Galois groups. Manuscripta Math. 43 (1983) 289–307

    Article  MathSciNet  MATH  Google Scholar 

  22. Girstmair, K.: On invariant polynomials and their application in field theory. Math. Compo 48 (1987) 781–797

    Article  MathSciNet  MATH  Google Scholar 

  23. Hulpke, A.: Block systems of a Galois group. Experimental Mathematics 4 (1995) 1 1–9

    MathSciNet  Google Scholar 

  24. Hulpke, A.: Konstruktion transitiver Permutationsgruppen. Ph.D. thesis, Rheinisch-Westflilische Technische Hochschule, Aachen, Germany, (1996)

    Google Scholar 

  25. Hulpke, A.: Galois groups through invariant relations. In Groups ’97 Bath/St. Andrews ( C. M. Campbell, E. F. Robertson, G. C. Smith, eds.). Cambridge University Press, to appear

    Google Scholar 

  26. Jordan, C.: Traite des substitutions et des equations algebriques. Gauthier-Villars, (1870)

    Google Scholar 

  27. Kliiners, J.: Uber die Berechnung von Automorphismen und Teilkorpern algebraischer Zahlkorper. Ph.D. thesis, Technische Universität, Berlin, (1997)

    Google Scholar 

  28. Kliiners, J., Pohst, M. E.: On computing subfields. J. Symb. Comput. 24 (1997) 385–397

    Article  Google Scholar 

  29. Krasner, M., Kaloujnine, L. A.: Produit complet des groupes de permutations et probleme d’extension de groupes II. Acta Sci. Math. (Szeged) 14 (1951) 39–66

    MathSciNet  MATH  Google Scholar 

  30. Küchlin, W. (ed.): Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation. The Association for Computing Machinery, ACM Press, (1997)

    Google Scholar 

  31. Lagarias, J. C., Odlyzko, A. M.: Effective versions of the Chebotarev density theorem. In Algebraic Number Fields (L-functions and Galois properties) (A. Frohlich, ed.). Academic Press, 1977 409–464

    Google Scholar 

  32. Landau, S.: How to tangle with a nested radical. Math. Intelligencer 16 (1994) 2 49–55

    Article  Google Scholar 

  33. Landau, S., Miller, G.: Solvability by radical is in polynomial time. J. Comput. System Sci. 30 (1985) 179–208 Techniques for the Computation of Galois Groups 77

    Article  MathSciNet  MATH  Google Scholar 

  34. Lazard, D., Valibouze, A.: Computing subfields: Reverse of the primitive element problem. In Computational Algebraic Geometry (F. Eyssette, A. Galligo, eds.), volume 109 of Progress in Mathematics. Birkhiiuser, Boston, 1993 163–176

    Chapter  Google Scholar 

  35. Lehobey, F.: Resolvent computations by resultants without extraneous powers. In Küchlin [30], 1997 85–92

    MathSciNet  Google Scholar 

  36. Lenstra, A., Lenstra, H., Lovcisz, L.: Factoring polynomials with rational coefficients. Math. Ann. 261 (1982) 515–534

    Article  MathSciNet  MATH  Google Scholar 

  37. Liebeck, M. W., Praeger, C. E., Saxl, J.: A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111 (1987) 365–383

    Article  MathSciNet  MATH  Google Scholar 

  38. Mattman, T. W., McKay, J.: Computation of Galois groups over function fields. Math. Compo 66 (1997) 218 823–831

    Article  MathSciNet  MATH  Google Scholar 

  39. Matzat, B. H.: Fortschritte in der inversen Galoistheorie. This volume

    Google Scholar 

  40. McKay, J., Stauduhar, R.: Finding relations among the roots of an irreducible polynomial. In Küchlin [30], 199775–77

    MathSciNet  Google Scholar 

  41. Mignotte, M.: Mathematics for Computer Algebra. Springer, Heidelberg, (1992)

    MATH  Google Scholar 

  42. Neukirch, J.: Algebraische Zahlentheorie. Springer, Heidelberg, (1992)

    MATH  Google Scholar 

  43. Schönert, M., et al.: GAP 3.4, patchlevel 4. Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, (1997)

    Google Scholar 

  44. Soicher, L. H.: The computation of Galois groups. Master’s thesis, Concordia University, (1981)

    Google Scholar 

  45. Soicher, L. H., McKay, J.: Computing Galois groups over the rationals. J. Number Theory 20 (1985) 273–281

    Article  MathSciNet  MATH  Google Scholar 

  46. Stauduhar, R. P.: The determination of Galois groups. Math. Compo 27 (1973) 981–996

    Article  MathSciNet  MATH  Google Scholar 

  47. Tschebotareff, N.: Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse geh6ren. Math. Ann. 95 (1925) 191–228

    Article  MathSciNet  MATH  Google Scholar 

  48. van der Waerden, B. L.: Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109 (1934) 13–1

    Article  MathSciNet  Google Scholar 

  49. van der Waerden, B. L.: Algebra, erster Teil. Springer, Heidelberg, eighth edition, (1971)

    Google Scholar 

  50. Wielandt, H.: Permutation groups through invariant relations and invariant functions. Lecture notes, Department of Mathematics, The Ohio State University, (1969)

    Google Scholar 

  51. Yokoyama, K: A modular method for computing the Galois group of polynomials. In Cohen and Roy [8], 1997 617–636

    MathSciNet  Google Scholar 

  52. Zassenhaus, H.: On Hensel factorization I. J. Number Theory 1 (1969) 291–311ss

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical and Computational Sciences, The University of St. Andrews, The North Haugh,St. Andrews, KY16 9SS, Fife, UK

    Alexander Hulpke

Authors
  1. Alexander Hulpke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, der Universität Heidelberg, Im Neuenheimer Feld 368, D-69120, Heidelberg, Germany

    B. Heinrich Matzat

  2. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Gert-Martin Greuel

  3. Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062, Aachen, Germany

    Gerhard Hiss

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hulpke, A. (1999). Techniques for the Computation of Galois Groups. In: Matzat, B.H., Greuel, GM., Hiss, G. (eds) Algorithmic Algebra and Number Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59932-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59932-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64670-9

  • Online ISBN: 978-3-642-59932-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation