Israel Journal of Mathematics

, Volume 109, Issue 1, pp 319–337 | Cite as

Hilbert’s irreducibility theorem for prime degree and general polynomials



Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt 0εℤ such thatf (X, t 0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t 0) is irreducible for all but finitely manyt 0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x 0,t 0) withx 0εℚ,t 0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.


Symmetric Group Maximal Subgroup Galois Group Permutation Group Irreducible Polynomial 


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Copyright information

© The Magnes Press 1999

Authors and Affiliations

  1. 1.IWR, Universität Heidelberg, Im Neuenheimer Feld 368HeidelbergGermany

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