On Hilbert's irreducibility theorem

2016/01/31 by Castillo, Abel, Dietmann, Rainer · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1602.00314

Abstract

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if f(X, T₁, …, T_s) is an irreducible polynomial with integer coefficients, having Galois group G over the function field ℚ(T₁, …, T_s), and K is any subgroup of G, then there are at most O_{f, ε}(H^s-1+|G/K|^-1+ε) specialisations t ∈ ℤ^s with |t| ≤ H such that the resulting polynomial f(X) has Galois group K over the rationals.

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• Hilbert properties of varieties

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• Effective Hilbert's Irreducibility Theorem for global fields
• Explicit Hilbert Irreducibility
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