Galois groups over rational function fields and Explicit Hilbert Irreducibility

Abstract

Let P∈Q[t,x] be a polynomial in two variables with rational coefficients, and let G be the Galois group of P over the field Q(t). It follows from Hilbert's Irreducibility Theorem that for most rational numbers c the specialized polynomial P(c,x) has Galois group isomorphic to G and factors in the same way as P. In this paper we discuss methods for computing the group G and obtaining an explicit description of the exceptional numbers c, i.e., those for which P(c,x) has Galois group different from G or factors differently from P. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.