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Israel Journal of Mathematics

, Volume 109, Issue 1, pp 319–337 | Cite as

Hilbert’s irreducibility theorem for prime degree and general polynomials

  • Peter MüllerEmail author
Article

Abstract

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt 0εℤ such thatf (X, t 0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t 0) is irreducible for all but finitely manyt 0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x 0,t 0) withx 0εℚ,t 0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.

Keywords

Symmetric Group Maximal Subgroup Galois Group Permutation Group Irreducible Polynomial 
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.

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References

  1. [Bil96]
    Y. Bilu,A note on universal Hilbert sets, Journal für die reine und angewandte Mathematik479 (1996), 195–203.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [DF96]
    P. Dèbes and M. Fried,Integral specialization of families of rational functions, preprint.Google Scholar
  3. [DM96]
    J. D. Dixon and B. Mortimer,Permutation Groups, Springer-Verlag, New York, 1996.zbMATHGoogle Scholar
  4. [DZ96]
    P. Dèbes and U. Zannier,Universal Hilbert subsets, Mathematical Proceedings of the Cambridge Philosophical Society124 (1998), 127–134.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Fri74]
    M. Fried,On Hilbert’s irreducibility theorem, Journal of Number Theory6 (1974), 211–231.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Fri94]
    M. Fried,Review of Serre’s ‘Topics in Galois Theory’, Bulletin of the American Mathematical Society (New Series)30(1) (1994), 124–135.MathSciNetGoogle Scholar
  7. [Fri95]
    M. Fried,Extension of constants, rigidity, and the Chowla-Zassenhaus conjecture, Finite Fields and their Application1 (1995), 326–359.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Gor68]
    D. Gorenstein,Finite Groups, Harper and Row, New York-Evanston-London, 1968.zbMATHGoogle Scholar
  9. [HB82]
    B. Huppert and N. Blackburn,Finite Groups III, Springer-Verlag, Berlin-Heidelberg, 1982.zbMATHGoogle Scholar
  10. [Hup67]
    B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg, 1967.zbMATHGoogle Scholar
  11. [Lan83]
    S. Lang,Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  12. [Mül96]
    P. Müller,Reducibility behavior of polynomials with varying coefficients, Israel Journal of Mathematics94 (1996), 59–91.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Neu72]
    P. M. Neumann,Transitive permutation groups of prime degree, Journal of the London Mathematical Society (2)5 (1972), 202–208.zbMATHCrossRefGoogle Scholar
  14. [Ser92]
    J.-P. Serre,Topics in Galois Theory, Jones and Bartlett, Boston, 1992.zbMATHGoogle Scholar
  15. [Shi74]
    K. Shih,On the construction of Galois extensions of function fields and number fields, Mathematische Annalen207 (1974), 99–120.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Sie29]
    C. L. Siegel,Über einige Anwendungen diophantischer Approximationen, Abhandlungen der Preussischen Akademie der Wissenschaften1 (1929), 41–69 (= Gesammelte Abhandlungen I, 209–266).Google Scholar
  17. [Völ96]
    H. Völklein,Groups as Galois Groups—An Introduction, Cambridge University Press, New York, 1996.zbMATHGoogle Scholar
  18. [Wie64]
    H. Wielandt,Finite Permutation Groups, Academic Press, New York-London, 1964.zbMATHGoogle Scholar
  19. [Zan96]
    U. Zannier,Note on dense Hilbert sets, Comptes Rendus de l’Académie des Sciences, Paris, Série I Mathématiques322 (1996), 703–706.zbMATHMathSciNetGoogle Scholar

Copyright information

© The Magnes Press 1999

Authors and Affiliations

  1. 1.IWR, Universität Heidelberg, Im Neuenheimer Feld 368HeidelbergGermany

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