Effective Hilbert’s irreducibility theorem for global fields

Abstract

We prove an effective form of Hilbert’s irreducibility theorem for polynomials over a global field K. More precisely, we give effective bounds for the number of specializations \(t \in {{\cal O}_K}\) that do not preserve the irreducibility or the Galois group of a given irreducible polynomial F(T, Y) ∈ K[T, Y]. The bounds are explicit in the height and degree of the polynomial F(T, Y), and are optimal in terms of the size of the parameter \(t \in {{\cal O}_K}\). Our proofs deal with the function field and number field cases in a unified way.