no. 4
Finiteness results for Hilbert's irreducibility theorem
[Résultats de finitude pour le théorème d'irréductibilité de Hilbert]
Müller, Peter1
1 Universität Heidelberg, IWR, Im Neuenheimer Feld 368, 69120 Heidelberg (Allemagne)
Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 983-1015.

Soient k un corps de nombres, 𝒪 k son anneau d’entiers et f(t,X)∈k(t)[X] un polynôme irréductible. Le théorème d’irréductibilité de Hilbert fournit une infinité de spécialisations entières t↦t ¯∈𝒪 k telles que f(t ¯,X) reste irréductible. Dans cet article, nous étudions l’ensemble Red f (𝒪 k ) des t ¯∈𝒪 k tels que f(t ¯,X) est réductible. Nous montrons que Red f (𝒪 k ) est un ensemble fini sous des hypothèses assez faibles. En particulier, certains de nos énoncés généralisent des résultats antérieurs obtenus par des techniques d’approximations diophantiennes. Notre méthode est différente. Nous utilisons de la théorie élémentaire des groupes, la théorie des valuations et le théorème de Siegel sur les points entiers des courbes algébriques. En utilisant en fait la généralisation de Siegel-Lang du théorème de Siegel, la plupart de nos résultats sont valables sur des corps assez généraux. On peut obtenir d’autres résultats en faisant appel à la classification des groupes finis simples. Nous en donnons un aperçu dans la dernière section.

Let k be a number field, 𝒪 k its ring of integers, and f(t,X)∈k(t)[X] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t↦t ¯∈𝒪 k such that f(t ¯,X) is still irreducible. In this paper we study the set Red f (𝒪 k ) of those t ¯∈𝒪 k with f(t ¯,X) reducible. We show that Red f (𝒪 k ) is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group theory, valuation theory, and Siegel’s theorem about integral points on algebraic curves. Indeed, using the Siegel-Lang extension of Siegel’s theorem, most of our results hold over more general fields. Using the classification of the finite simple groups, further results can be obtained. The last section contains a short survey.

DOI : 10.5802/aif.1907
Classification : 12E25, 12E30, 14H25, 20B15, 20B25
Keywords: Hilbert's irreducibility theorem, Hilbert sets, permutation groups
Mot clés : théorème d'irréductibilité de Hilbert, parties hilbertiennes, groupes de permutation

Müller, Peter 1

1 Universität Heidelberg, IWR, Im Neuenheimer Feld 368, 69120 Heidelberg (Allemagne) 
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Müller, Peter. Finiteness results for Hilbert's irreducibility theorem. Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 983-1015. doi : 10.5802/aif.1907. https://aif.centre-mersenne.org/articles/10.5802/aif.1907/

[Cav00] M. Cavachi On a special case of Hilbert's irreducibility theorem, J. Number Theory, Volume 82 (2000), pp. 96-99 | DOI | MR | Zbl

[Dèb86] P. Dèbes G-fonctions et théorème d`irréductibilité de Hilbert, Acta Arith, Volume 47 (1986), pp. 371-402 | EuDML | MR | Zbl

[Dèb92] P. Dèbes On the irreducibility of the polynomials P(t m ,Y), J. Number Theory, Volume 42 (1992), pp. 141-157 | DOI | MR | Zbl

[Dèb96] P. Dèbes Hilbert subsets and S-integral points, Manuscripta Math., Volume 89 (1996) no. 1, pp. 107-137 | DOI | EuDML | MR | Zbl

[DF99] P. Dèbes; M. D. Fried Integral specialization of families of rational functions, Pacific J. Math, Volume 190 (1999) no. 1, pp. 45-85 | DOI | MR | Zbl

[DM96] J. D. Dixon; B. Mortimer Permutation Groups, Springer-Verlag, New York, 1996 | MR | Zbl

[FM69] M. Fried; R. E. MacRae On the invariance of chains of fields, Illinois J. Math., Volume 13 (1969), pp. 165-171 | MR | Zbl

[Fri74] M. Fried On Hilbert's irreducibility theorem, J. Number Theory, Volume 6 (1974), pp. 211-231 | DOI | MR | Zbl

[Fri77] M. Fried Fields of definition of function fields and Hurwitz families -- Groups as Galois groups, Comm. Algebra, Volume 5 (1977), pp. 17-82 | DOI | MR | Zbl

[Fri80] M. Fried Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem, The Santa Cruz Conference on Finite Groups (Proc. Sympos. Pure Math.), Volume vol. 37 (1980), pp. 571-602 | Zbl

[Fri85] M. Fried On the Sprind\v zuk-Weissauer approach to universal Hilbert subsets, Israel J. Math., Volume 51 (1985) no. 4, pp. 347-363 | DOI | MR | Zbl

[Gor68] D. Gorenstein Finite Groups, Harper and Row, New York-Evanston-London, 1968 | MR | Zbl

[Gro71] A. Grothendieck Revêtement étales et groupe fondamental, SGA1 (Lecture Notes in Math.), Volume vol. 224 (1971)

[GT90] R. M. Guralnick; J. G. Thompson Finite groups of genus zero, J. Algebra, Volume 131 (1990), pp. 303-341 | DOI | MR | Zbl

[Gur00] R. M. Guralnick Monodromy groups of curves (Preprint)

[HB82] B. Huppert; N. Blackburn Finite Groups III, Springer-Verlag, Berlin Heidelberg, 1982 | MR | Zbl

[Isa76] I. M. Isaacs Character Theory of Finite Groups, Pure and Applied Mathematics, 69, Academic Press, 1976 | MR | Zbl

[Kli98] N. Klingen Arithmetical Similarities -- Prime Decomposition and Finite Group Theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1998 | MR | Zbl

[Lan00] K. Langmann Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien II, Math. Nachr., Volume 211 (2000), pp. 79-108 | DOI | MR | Zbl

[Lan83] S. Lang Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983 | MR | Zbl

[Lan90] K. Langmann Ganzalgebraische Punkte und der Hilbertsche Irreduzibilitätssatz, J. Reine Angew. Math, Volume 405 (1990), pp. 131-146 | DOI | MR | Zbl

[Lan94] K. Langmann Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien, Math. Ann, Volume 299 (1994), pp. 127-153 | DOI | MR | Zbl

[MM99] G. Malle; B. H. Matzat Inverse Galois Theory, Springer-Verlag, Berlin, 1999 | MR | Zbl

[Mül01] P. Müller Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Siegel functions (submitted)

[Mül99] P. Müller Hilbert's irreducibility theorem for prime degree and general polynomials, Israel J. Math, Volume 109 (1999), pp. 319-337 | DOI | MR | Zbl

[Sco77] L. Scott Matrices and cohomology, Anal. Math, Volume 105 (1977), pp. 473-492 | DOI | MR | Zbl

[Ser79] J.-P. Serre Local Fields, Springer-Verlag, New York, 1979 | MR | Zbl

[Sie29] C. L. Siegel Über einige Anwendungen diophantischer Approximationen (Ges. Abh., I), Abh. Pr. Akad. Wiss., Volume 1 (1929), p. 41-69 ; 209-266

[Spr83] V. G. Sprind{#x017E;}uk Arithmetic specializations in polynomials, J. Reine Angew. Math, Volume 340 (1983), pp. 26-52 | MR | Zbl

[Völ96] H. Völklein Groups as Galois Groups -- an Introduction, Cambridge University Press, New York, 1996 | MR | Zbl

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