Explicit Hilbert Irreducibility

TL;DR

This paper presents a method to explicitly identify the set of rational numbers for which a specialized polynomial either becomes reducible or changes its Galois group, extending Hilbert's Irreducibility Theorem.

Contribution

The paper introduces a novel explicit approach to determine exceptional specializations of polynomials, providing concrete examples for degrees four and six.

Findings

Explicit description of exceptional specializations for specific polynomials

Method to determine when polynomial specializations alter Galois groups

Application to polynomials of degrees four and six

Abstract

Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has the same Galois group as $P$. We discuss here a method for obtaining an explicit description of the set of exceptional numbers $t$, i.e., those for which $P(t,X)$ is either reducible or has a different Galois group than $P$. To illustrate the method we determine the exceptional specializations of two polynomials of degrees four and six.