Soient un corps de nombres, son anneau d’entiers et un polynôme irréductible. Le théorème d’irréductibilité de Hilbert fournit une infinité de spécialisations entières telles que reste irréductible. Dans cet article, nous étudions l’ensemble des tels que est réductible. Nous montrons que 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 be a number field, its ring of integers, and be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations such that is still irreducible. In this paper we study the set of those with reducible. We show that 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.
Classification : 12E25, 12E30, 14H25, 20B15, 20B25
Mots clés : théorème d'irréductibilité de Hilbert, parties hilbertiennes, groupes de permutation
@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}, pages = {983--1015}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {4}, year = {2002}, doi = {10.5802/aif.1907}, zbl = {1014.12002}, mrnumber = {1926669}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1907/} }
TY - JOUR AU - Müller, Peter TI - Finiteness results for Hilbert's irreducibility theorem JO - Annales de l'Institut Fourier PY - 2002 DA - 2002/// SP - 983 EP - 1015 VL - 52 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1907/ UR - https://zbmath.org/?q=an%3A1014.12002 UR - https://www.ams.org/mathscinet-getitem?mr=1926669 UR - https://doi.org/10.5802/aif.1907 DO - 10.5802/aif.1907 LA - en ID - AIF_2002__52_4_983_0 ER -
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. http://www.numdam.org/articles/10.5802/aif.1907/
[Cav00] On a special case of Hilbert's irreducibility theorem, J. Number Theory, Volume 82 (2000), pp. 96-99 | Article | MR 1755156 | Zbl 0985.12001
[Dèb86] G-fonctions et théorème d`irréductibilité de Hilbert, Acta Arith, Volume 47 (1986), pp. 371-402 | EuDML 206037 | MR 884733 | Zbl 0565.12012
[Dèb92] On the irreducibility of the polynomials , J. Number Theory, Volume 42 (1992), pp. 141-157 | Article | MR 1183373 | Zbl 0770.12005
[Dèb96] Hilbert subsets and S-integral points, Manuscripta Math., Volume 89 (1996) no. 1, pp. 107-137 | Article | EuDML 156154 | MR 1368540 | Zbl 0853.12001
[DF99] Integral specialization of families of rational functions, Pacific J. Math, Volume 190 (1999) no. 1, pp. 45-85 | Article | MR 1722766 | Zbl 1016.12002
[DM96] Permutation Groups, Springer-Verlag, New York, 1996 | MR 1409812 | Zbl 0951.20001
[FM69] On the invariance of chains of fields, Illinois J. Math., Volume 13 (1969), pp. 165-171 | MR 238815 | Zbl 0174.07302
[Fri74] On Hilbert's irreducibility theorem, J. Number Theory, Volume 6 (1974), pp. 211-231 | Article | MR 349624 | Zbl 0299.12002
[Fri77] Fields of definition of function fields and Hurwitz families -- Groups as Galois groups, Comm. Algebra, Volume 5 (1977), pp. 17-82 | Article | MR 453746 | Zbl 0478.12006
[Fri80] 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 0451.14011
[Fri85] On the Sprind\v zuk-Weissauer approach to universal Hilbert subsets, Israel J. Math., Volume 51 (1985) no. 4, pp. 347-363 | Article | MR 804491 | Zbl 0579.12002
[Gor68] Finite Groups, Harper and Row, New York-Evanston-London, 1968 | MR 231903 | Zbl 0185.05701
[Gro71] Revêtement étales et groupe fondamental, SGA1 (Lecture Notes in Math.), Volume vol. 224 (1971)
[GT90] Finite groups of genus zero, J. Algebra, Volume 131 (1990), pp. 303-341 | Article | MR 1055011 | Zbl 0713.20011
[Gur00] Monodromy groups of curves (Preprint)
[HB82] Finite Groups III, Springer-Verlag, Berlin Heidelberg, 1982 | MR 662826 | Zbl 0514.20002
[Isa76] Character Theory of Finite Groups, Pure and Applied Mathematics, 69, Academic Press, 1976 | MR 460423 | Zbl 0337.20005
[Kli98] Arithmetical Similarities -- Prime Decomposition and Finite Group Theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1998 | MR 1638821 | Zbl 0896.11042
[Lan00] Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien II, Math. Nachr., Volume 211 (2000), pp. 79-108 | Article | MR 1743486 | Zbl 0995.11044
[Lan83] Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983 | MR 715605 | Zbl 0528.14013
[Lan90] Ganzalgebraische Punkte und der Hilbertsche Irreduzibilitätssatz, J. Reine Angew. Math, Volume 405 (1990), pp. 131-146 | Article | MR 1040999 | Zbl 0687.14001
[Lan94] Werteverhalten holomorpher Funktionen auf Überlagerungen und zahlentheoretische Analogien, Math. Ann, Volume 299 (1994), pp. 127-153 | Article | MR 1273080 | Zbl 0805.11077
[MM99] Inverse Galois Theory, Springer-Verlag, Berlin, 1999 | MR 1711577 | Zbl 0940.12001
[Mül01] Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Siegel functions (submitted)
[Mül99] Hilbert's irreducibility theorem for prime degree and general polynomials, Israel J. Math, Volume 109 (1999), pp. 319-337 | Article | MR 1679603 | Zbl 0926.12001
[Sco77] Matrices and cohomology, Anal. Math, Volume 105 (1977), pp. 473-492 | Article | MR 447434 | Zbl 0399.20047
[Ser79] Local Fields, Springer-Verlag, New York, 1979 | MR 554237 | Zbl 0423.12016
[Sie29] Über einige Anwendungen diophantischer Approximationen (Ges. Abh., I), Abh. Pr. Akad. Wiss., Volume 1 (1929), p. 41-69 ; 209-266
[Spr83] Arithmetic specializations in polynomials, J. Reine Angew. Math, Volume 340 (1983), pp. 26-52 | MR 691959 | Zbl 0497.12001
[Völ96] Groups as Galois Groups -- an Introduction, Cambridge University Press, New York, 1996 | MR 1405612 | Zbl 0868.12003
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