Galois groups over rational function fields and Explicit Hilbert Irreducibility. (March 2021) - Krumm, David; Sutherland, Nicole
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.