Advertisement

Rationality Criteria for Galois Extensions

  • B. Heinrich Matzat
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 16)

Abstract

Some rationality criteria for finite Galois extensions overℂ(t)are explained. The first rationality criterion and the second rationality criterion, together with the corresponding examples are contained in the forthcoming lecture notes [27] (see also [23—25]). The rationality criteria in sections 4 and 5, the braid orbit theorem, and the twisted braid orbit theorem, are new. With the last one, the Mathieu group M24 is realized as Galois group over ℚ.

Keywords

Finite Group Conjugacy Class Galois Group Class Structure Class 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.
    M. Aschbacher, D. Gorenstein, R. Lyons, M. O’Nan, C. Sims and W. Feit, eds., “Proceedings of the Rutgers group theory year, 1983–1984,” Cambridge University Press, 1984.MATHGoogle Scholar
  2. 2.
    G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), 267–276; Math. USSR Izv. 14 (1980), 247–256.MathSciNetMATHGoogle Scholar
  3. 3.
    G. V. Belyi, On extensions of the maximal cyclotomic field having a given classical Galois group, J. reine angew. Math. 341 (1983), 147–156.MathSciNetMATHGoogle Scholar
  4. 4.
    R. Biggers and M. D. Fried, Moduli spaces of covers and the Hurwitz monodromy group, J. reine angew. Math. 335 (1982), 87–121.MathSciNetMATHGoogle Scholar
  5. 5.
    J. S. Birman, “Braids, Links, and Mapping Class Groups,” Princeton University Press, 1974.Google Scholar
  6. 6.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, “Atlas of finite groups,” Clarendon Press, Oxford, 1985.MATHGoogle Scholar
  7. 7.
    R. Dentzer, Projektive symplektische Gruppen PSp$(p) als Galoisgruppen über Q(t), Arch. Math, (to appear).Google Scholar
  8. 8.
    W. Feit and P. Fong, Rational rigidity of G2(p) for any prime p gt 5, in “Proceedings of the Rutgers group theory year, 1983–1984, ” Cambridge University Press, 1984, pp. 323–326.Google Scholar
  9. 9.
    M. D. Fried, Fields of definition of function fields and Hurwitz families — Groups as Galois groups, Commun. Alg. 5 (1977), 17–82.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    M. D. Fried, On reduction of the inverse Galois group problem to simple groups, in “Proceedings of the Rutgers group theory year, 1983–1984, ” Cambridge University Press, 1984, pp. 2889–301.Google Scholar
  11. 11.
    M. D. Fried and M. Jarden, “Field Arithmetic,” Springer-Verlag, 1986.Google Scholar
  12. 12.
    D. Hilbert, Uber die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten, J. reine angew. Math. 110 (1892), 104–129.Google Scholar
  13. 13.
    G. Hoyden-Siedersleben, Realisierung der Jankogurppen J\ und J2 als Galoisgruppen über Q, J. Algebra 97 (1985), 14–22.MathSciNetCrossRefGoogle Scholar
  14. 14.
    G. Hoyden-Siedersleben and B. H. Matzat, Realisierung sporadischer einfacher Gruppen als Galoisgruppen über Kreisteilungskörpem, J. Algebra 101 (1986), 273–285.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D. C. Hunt, Rational rigidity and the sporadic groups, J. Algebra 99 (1986), 577–592.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. Hurwitz, Uber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1–61.MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. E. LaMacchia, Polynomials with Galois group PSL(2,7), Commun. Algebra 8 (1980), 983–992.CrossRefGoogle Scholar
  18. 18.
    G. Malle, Polynomials for primitive nonsolvable permutation groups of degree d lt 15, J. Symb. Comp. 4 (1987), 83–92.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    G. Malle, Polynomials with Galois groups Aut(M22gt M22 and over Q, Math. Comp, (to appear).Google Scholar
  20. 20.
    G. Malle, Exceptional groups of Lie type as Galois groups, J. reine angew. Math, (to appear).Google Scholar
  21. 21.
    G. Malle and B. H. Matzat, Realisierung von Gruppen PSL2(Fp) als Galoisgruppen über Q, Math. Ann. 272 (1985), 5489–565.MathSciNetGoogle Scholar
  22. 22.
    B. H. Matzat, Konstruktion von Zahlkörpern mit der Galoisgruppe Mi 2 über Q(\/—5), Arch. Math. 40 (1983), 245–254.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    B. H. Matzat, Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe, J. reine angew. Math. 349 (1984), 1789–220.MathSciNetGoogle Scholar
  24. 24.
    B. H. Matzat, Zwei Aspekte konstruktiver Galoistheorie, J. Algebra 96 (1985), 4989–531.MathSciNetCrossRefGoogle Scholar
  25. 25.
    B. H. Matzat, Topologische Automorphismen in der konstruktiven Galoistheorie, J. reine angew. Math. 371 (1986), 16–45.MathSciNetMATHGoogle Scholar
  26. 26.
    B. H. Matzat, Uber das Umkehrproblem der Galoisschen Theorie, Jber. Deutsch. Math.-Ver. (to appear).Google Scholar
  27. 27.
    B. H. Matzat, “Konstruktive Galoistheorie,” Springer-Verlag, 1987.Google Scholar
  28. 28.
    B. H. Matzat, Computational methods in constructive Galois theory, in “Trends in Computer Algebra,” R. Janßen, ed., Springer-Verlag, 1988, pp. 137–155.CrossRefGoogle Scholar
  29. 29.
    B. H. Matzat, Zöpfe und Galoissche Gruppen, in preparation.Google Scholar
  30. 30.
    B. H. Matzat and A. Zeh-Marschke, Realisierung der Mathieugruppen M\\ und Mi2 als Galoisgruppen über Q, J. Number Theory 23 (1986), 195–202.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    B. H. Matzat and A. Zeh-Marschke, Polynome mit der Galoisgruppe M\\ über Q, J. Symb. Comp. 4 (1987), 93–97.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    H. Pahlings, Some sporadic groups as Galois groups, Rend. Sem. Math. Univ. Padova (to appear).Google Scholar
  33. 33.
    K. A. Ribet, On l-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    I. R. Safarevic, Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR, Ser. Mat. 18 (1954), 525–578; Amer. Math. Soc. Transl. 4 (1956), 185–237.Google Scholar
  35. 35.
    J.-P. Serre, “Groupes algebriques et corps de classes,” Hermann, Paris, 1959.Google Scholar
  36. 36.
    K.-y. Shih, On the construction of Galois extensions of function fields and number fields, Math. Ann. 207 (1974), 989–120.Google Scholar
  37. 37.
    J. G. Thompson, Some finite groups which appear as Ga\(L/K), where K lt Q(/in), J. Algebra 89 (1984), 437–499.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    J. G. Thompson, PSL3 and Galois groups over Q, in “Proceedings of the Rutgers group theory year, 1983–1984, ” Cambridge University Press, 1984, pp. 3089–319.Google Scholar
  39. 39.
    J. G. Thompson, Rational rigidity o/G2(5), in “Proceedings of the Rutgers group theory year, 1983–1984, ” Cambridge University Press, 1984, pp. 321–322.Google Scholar
  40. 40.
    J. G. Thompson, Primitive roots and rigidity, in “Proceedings of the Rutgers group theory year, 1983–1984, ” Cambridge University Press, 1984, pp. 327–350.Google Scholar
  41. 41.
    H. Weber, “Lehrbuch der Algebra III,” Vieweg, Braunschweig, 1908.Google Scholar
  42. 42.
    R. Weissauer, Der Hilbertsche Irreduzibilitätssatz, J. reine angew. Math. 334 (1982), 203–220.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • B. Heinrich Matzat
    • 1
  1. 1.Mathematisches Institut IIUniversität Karlsruhe (TH)Karlsruhe 1Federal Republic of Germany

Personalised recommendations