The least admissible value of the parameter in Hilbert's Irreducibility Theorem
Andrzej Schinzel, Umberto M. Zannier · Acta Arithmetica · 1995
The simplest case of Hilbert’s Irreducibility Theorem asserts that if F (t, x) is irreducible over Q, then there exists t∗ ∈ Q such that F (t∗, x) is irreducible over Q. Many different proofs have been given for this theorem, namely Hilbert’s (1892) [H], Mertens’s (1911) [Me], Skolem’s (1921) [Sk], Dorge’s (1927) [Do], Siegel’s (1929) [Si], Eichler’s (1939) [Ei], Inaba’s (1943) [In], Fried’s (1974) [Fr], Roquette’s (1975) [Ro], Cohen’s (1981) [Co], Sprindžuk’s (1981) [Spr], Debes’s (1986) [De1], (1993) [De2]. Only the last of the quoted papers explicitly mentions the problem of estimating the size of a t∗ with the above property in terms of the degree and height of F . By the height of F , to be abbreviated H(F ), we mean the maximum absolute value of the coefficients of a constant multiple of F that has coprime integer coefficients. Debes gives actually an estimate valid for several polynomials Fi. His result reads (see Cor. 3.7 of [De2]): Let F1, . . . , Fh be irreducible polynomials in Q[t, x] such that degFi ≤ D and H(Fi) ≤ H (1). Then there exists a rational number t∗ = u/v such that each Fi(t∗, x) is irreducible over Q and