Hilbert’s irreducibility theorem for prime degree and general polynomials |
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Authors: | Peter Müller |
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Institution: | 1. IWR, Universit?t Heidelberg, Im Neuenheimer Feld 368, D-69120, Heidelberg, Germany
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Abstract: | Letf (X, t)εℚX, 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.
Supported by the DFG. |
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Keywords: | |
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