Finiteness results for Hilbert's irreducibility theorem
Annales de l'Institut Fourier, Volume 52 (2002) no. 4, p. 983-1015

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 tt ¯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.

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 tt ¯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.

DOI : https://doi.org/10.5802/aif.1907
Classification:  12E25,  12E30,  14H25,  20B15,  20B25
Keywords: Hilbert's irreducibility theorem, Hilbert sets, permutation groups
@article{AIF_2002__52_4_983_0,
     author = {M\"uller, Peter},
     title = {Finiteness results for Hilbert's irreducibility theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     pages = {983-1015},
     doi = {10.5802/aif.1907},
     zbl = {1014.12002},
     mrnumber = {1926669},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_4_983_0}
}
Müller, Peter. Finiteness results for Hilbert's irreducibility theorem. Annales de l'Institut Fourier, Volume 52 (2002) no. 4, pp. 983-1015. doi : 10.5802/aif.1907. http://www.numdam.org/item/AIF_2002__52_4_983_0/

[Cav00] M. Cavachi On a special case of Hilbert's irreducibility theorem, J. Number Theory, Tome 82 (2000), pp. 96-99 | Article | MR 1755156 | Zbl 0985.12001

[Dèb86] P. Dèbes G-fonctions et théorème d`irréductibilité de Hilbert, Acta Arith, Tome 47 (1986), pp. 371-402 | MR 884733 | Zbl 0565.12012

[Dèb92] P. Dèbes On the irreducibility of the polynomials P(t m ,Y), J. Number Theory, Tome 42 (1992), pp. 141-157 | Article | MR 1183373 | Zbl 0770.12005

[Dèb96] P. Dèbes Hilbert subsets and S-integral points, Manuscripta Math., Tome 89 (1996) no. 1, pp. 107-137 | Article | MR 1368540 | Zbl 0853.12001

[DF99] P. Dèbes; M. D. Fried Integral specialization of families of rational functions, Pacific J. Math, Tome 190 (1999) no. 1, pp. 45-85 | Article | MR 1722766 | Zbl 1016.12002

[DM96] J. D. Dixon; B. Mortimer Permutation Groups, Springer-Verlag, New York (1996) | MR 1409812 | Zbl 0951.20001

[FM69] M. Fried; R. E. Macrae On the invariance of chains of fields, Illinois J. Math., Tome 13 (1969), pp. 165-171 | MR 238815 | Zbl 0174.07302

[Fri74] M. Fried On Hilbert's irreducibility theorem, J. Number Theory, Tome 6 (1974), pp. 211-231 | Article | MR 349624 | Zbl 0299.12002

[Fri77] M. Fried Fields of definition of function fields and Hurwitz families -- Groups as Galois groups, Comm. Algebra, Tome 5 (1977), pp. 17-82 | Article | MR 453746 | Zbl 0478.12006

[Fri80] M. Fried Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem, The Santa Cruz Conference on Finite Groups, Amer. Math. Soc., Providence, Rhode Island (Proc. Sympos. Pure Math.) Tome vol. 37 (1980), pp. 571-602 | Zbl 0451.14011

[Fri85] M. Fried On the Sprind\v zuk-Weissauer approach to universal Hilbert subsets, Israel J. Math., Tome 51 (1985) no. 4, pp. 347-363 | Article | MR 804491 | Zbl 0579.12002

[Gor68] D. Gorenstein Finite Groups, Harper and Row, New York-Evanston-London (1968) | MR 231903 | Zbl 0185.05701

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

[GT90] R. M. Guralnick; J. G. Thompson Finite groups of genus zero, J. Algebra, Tome 131 (1990), pp. 303-341 | Article | MR 1055011 | Zbl 0713.20011

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

[HB82] B. Huppert; N. Blackburn Finite Groups III, Springer-Verlag, Berlin Heidelberg (1982) | MR 662826 | Zbl 0514.20002

[Isa76] I. M. Isaacs Character Theory of Finite Groups, Academic Press, Pure and Applied Mathematics, Tome 69 (1976) | MR 460423 | Zbl 0337.20005

[Kli98] N. Klingen Arithmetical Similarities -- Prime Decomposition and Finite Group Theory, Oxford University Press, Oxford, Oxford Mathematical Monographs (1998) | MR 1638821 | Zbl 0896.11042

[Lan00] K. Langmann Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien II, Math. Nachr., Tome 211 (2000), pp. 79-108 | Article | MR 1743486 | Zbl 0995.11044

[Lan83] S. Lang Fundamentals of Diophantine Geometry, Springer-Verlag, New York (1983) | MR 715605 | Zbl 0528.14013

[Lan90] K. Langmann Ganzalgebraische Punkte und der Hilbertsche Irreduzibilitätssatz, J. Reine Angew. Math, Tome 405 (1990), pp. 131-146 | Article | MR 1040999 | Zbl 0687.14001

[Lan94] K. Langmann Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien, Math. Ann, Tome 299 (1994), pp. 127-153 | Article | MR 1273080 | Zbl 0805.11077

[MM99] G. Malle; B. H. Matzat Inverse Galois Theory, Springer-Verlag, Berlin (1999) | MR 1711577 | Zbl 0940.12001

[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, Tome 109 (1999), pp. 319-337 | Article | MR 1679603 | Zbl 0926.12001

[Sco77] L. Scott Matrices and cohomology, Anal. Math, Tome 105 (1977), pp. 473-492 | Article | MR 447434 | Zbl 0399.20047

[Ser79] J.-P. Serre Local Fields, Springer-Verlag, New York (1979) | MR 554237 | Zbl 0423.12016

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

[Spr83] V. G. Sprind{#X017E;}Uk Arithmetic specializations in polynomials, J. Reine Angew. Math, Tome 340 (1983), pp. 26-52 | MR 691959 | Zbl 0497.12001

[Völ96] H. Völklein Groups as Galois Groups -- an Introduction, Cambridge University Press, New York (1996) | MR 1405612 | Zbl 0868.12003