Galois groups over rational function fields and Explicit Hilbert Irreducibility. (March 2021)

• Record Type: Journal Article
• Authors: Krumm, David; Sutherland, Nicole
• Is Part Of: Journal of symbolic computation. Volume 103(2021)
• Page Start: 108
• Page End: 126
• Publication Date: 2021-03
• Subjects: 11R32 -- 11R09; Hilbert Irreducibility Theorem -- Galois groups -- Function fields; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.11.003
• Languages: English
• ISSNs: 0747-7171

Abstract

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.