On Hilbert’s irreducibility theorem

Abel Castillo¹ ; Rainer Dietmann²

¹ Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 S Morgan St Chicago, IL 60607, U.S.A.
² Department of Mathematics Royal Holloway University of London TW20 0EX Egham, United Kingdom

Acta Arithmetica, Tome 180 (2017) no. 1, pp. 1-14.

Résumé

We obtain new quantitative forms of Hilbert’s irreducibility theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb Q(T_1, \ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$ specialisations $\mathbf{t} \in \mathbb Z^s$ with $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.

Détail

DOI : 10.4064/aa8380-2-2017

Keywords: obtain quantitative forms hilbert irreducibility theorem particular ldots irreducible polynomial integer coefficients having galois group function field mathbb ldots subgroup there varepsilon s varepsilon specialisations mathbf mathbb mathbf resulting polynomial has galois group rationals