Israel Journal of Mathematics

, Volume 94, Issue 1, pp 59–91 | Cite as

Reducibility behavior of polynomials with varying coefficients

  • Peter Müller


LetK be a number field, and lethK[Y] be a polynomial of degreen. Fix an integer 0≤i≤n and compare the set ν of those integersa ofK such thath(Y)−aY ihas a root inK with the set\(\mathcal{R}\) of those integersa, such thath(Y)−aY iis reducible overK. Ifi is coprime ton, then we classify the rare cases where ν is not cofinite in\(\mathcal{R}\). The main tools are a theorem of Siegel about integral points on algebraic curves and the theory of finite groups.


Normal Subgroup Finite Group Permutation Group Number Field Algebraic Closure 
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  1. [1]
    M. Aschbacher,\(\mathcal{F} - sets\)-sets and permutation groups, Journal of Algebra30 (1974), 400–416.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. J. Cameron,Finite permutation groups and finite simple groups, The Bulletin of the London Mathematical Society13 (1981), 1–22.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    C. Chevalley,Algebraic Functions of One Variable, Mathematical Surveys VI, American Mathematical Society, Providence, 1951.zbMATHGoogle Scholar
  4. [4]
    J. Conway, R. Curtis, S. Norton, R. Parker and R. Wilson,Atlas of Finite Groups Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford, New York, 1985.zbMATHGoogle Scholar
  5. [5]
    P. Dèbes,G-fonctions et théorème d'irréductibilité de Hilbert, Acta Arithmetica47 (1986), 371–402.zbMATHMathSciNetGoogle Scholar
  6. [6]
    P. Dèbes,On the irreducibility of the polynomials P(tm, Y), Journal of Number Theory42 (1992), 141–157.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    M. Fried,The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois Journal of Mathematics,17 (1973), 128–146.zbMATHMathSciNetGoogle Scholar
  8. [8]
    M. Fried,On Hilbert's Irreducibility Theorem, Journal of Number Theory6 (1974), 211–231.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Fried,Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem, The Santa Cruz conference on finite groups, Proc. Symp. Pure Math., Vol. 37, American Mathematical Society, Providence, R.I., 1980, pp. 571–602.Google Scholar
  10. [10]
    M. Fried,Rigidity and applications of the classification of simple groups to monodromy, Part II—Applications of connectivity; Davenport and Hilbert-Siegel Problems, preprint.Google Scholar
  11. [11]
    M. Fried,Review of Serre's ‘Topics in Galois Theory’, Bulletin of the American Mathematical Society30 (1994), 124–135.Google Scholar
  12. [12]
    M. Fried and M. Jarden,Field Arithmetic, Springer, Berlin-Heidelberg, 1986.zbMATHGoogle Scholar
  13. [13]
    D. Gorenstein,Finite Groups, Harper and Row, New York-Evanston-London, 1968.zbMATHGoogle Scholar
  14. [14]
    R. M. Guralnick and J. G. Thompson,Finite groups of genus zero, Journal of Algebra131 (1990), 303–341.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    B. Huppert,Endliche Gruppen I, Springer, Berlin, 1983.Google Scholar
  16. [16]
    W. M. Kantor,Linear groups containing a Singer cycle, Journal of Algebra62 (1980), 232–234.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Lang,Algebra, Addison-Wesley, Reading, 1984.zbMATHGoogle Scholar
  18. [18]
    S. Lang,Fundamentals of Diophantine Geometry, Springer, New York, 1983.zbMATHGoogle Scholar
  19. [19]
    S. Lang,Algebraic Number Theory, Springer, New York, 1986.zbMATHGoogle Scholar
  20. [20]
    B. H. Matzat,Konstruktive Galoistheorie, Lecture Notes in Mathematics1284, Springer, Berlin, 1987.zbMATHGoogle Scholar
  21. [21]
    P. Müller,Monodromiegruppen rationaler Funktionen und Polynome mit variablen Koeffizienten, Thesis, 1994.Google Scholar
  22. [22]
    M. Neubauer,On primitive monodromy groups of genus zero and one, I, Communications in Algebra21 (1993), 711–746.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    M. Neubauer,On primitive monodromy groups of genus zero and one, II, preprint.Google Scholar
  24. [24]
    M. E. O'Nan,Normal structure of the one-point stabilizer of a doubly-transitive permutation group. II, Transactions of the American Mathematical Society214 (1975), 43–74.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    J.-P. Serre,Topics in Galois Theory, Jones and Bartlett, Boston, 1992.zbMATHGoogle Scholar
  26. [26]
    C. L. Siegel,Über einige Anwendungen diophantischer Approximationen, Abh. Pr. Akad. Wiss.1 (1929), 41–69 (=Ges. Abh., I, 209–266).Google Scholar
  27. [27]
    H. Wielandt,Finite Permutation Groups, Academic Press, New York and London, 1964.zbMATHGoogle Scholar

Copyright information

© The Magnes Press 1996

Authors and Affiliations

  • Peter Müller
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

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