Finiteness results for Hilbert's irreducibility theorem

Peter Müller[1]

  • [1] Universität Heidelberg, IWR, Im Neuenheimer Feld 368, 69120 Heidelberg (Allemagne)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 4, page 983-1015
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Müller, Peter. "Finiteness results for Hilbert's irreducibility theorem." Annales de l’institut Fourier 52.4 (2002): 983-1015. <http://eudml.org/doc/116011>.

@article{Müller2002,
abstract = {Let $k$ be a number field, $\{\mathcal \{O\}\}_k$ its ring of integers, and $f(t,X)\in k(t)[X]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \bar\{t\}\in \{\mathcal \{O\}\}_k$ such that $f(\bar\{t\},X)$ is still irreducible. In this paper we study the set $\{\rm Red\}_f(\{\mathcal \{O\}\}_k)$ of those $\bar\{t\}\in \{\mathcal \{O\}\}_k$ with $f(\bar\{t\},X)$ reducible. We show that $\{\rm Red\}_f(\{\mathcal \{O\}\}_k)$ 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.},
affiliation = {Universität Heidelberg, IWR, Im Neuenheimer Feld 368, 69120 Heidelberg (Allemagne)},
author = {Müller, Peter},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert's irreducibility theorem; Hilbert sets; permutation groups},
language = {eng},
number = {4},
pages = {983-1015},
publisher = {Association des Annales de l'Institut Fourier},
title = {Finiteness results for Hilbert's irreducibility theorem},
url = {http://eudml.org/doc/116011},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Müller, Peter
TI - Finiteness results for Hilbert's irreducibility theorem
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 983
EP - 1015
AB - Let $k$ be a number field, ${\mathcal {O}}_k$ its ring of integers, and $f(t,X)\in k(t)[X]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \bar{t}\in {\mathcal {O}}_k$ such that $f(\bar{t},X)$ is still irreducible. In this paper we study the set ${\rm Red}_f({\mathcal {O}}_k)$ of those $\bar{t}\in {\mathcal {O}}_k$ with $f(\bar{t},X)$ reducible. We show that ${\rm Red}_f({\mathcal {O}}_k)$ 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.
LA - eng
KW - Hilbert's irreducibility theorem; Hilbert sets; permutation groups
UR - http://eudml.org/doc/116011
ER -

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