Reducible specializations of polynomials: the nonsolvable case
Joachim König, Danny Neftin · arXiv (Cornell University) · 2020
Given an irreducible polynomial F in Q(t)[x], we develop methods for determining the set of exceptions in Hilbert's irreducibility theorem up to a finite set, under nonsolvability assumptions on its Galois group A=Gal(F/Q(t)). As opposed to previous results, these methods address the case where A is imprimitive. As a consequence, we answer the Davenport-Lewis-Schinzel problem (1959), and the problem of determining the reducibility of fibers of a polynomial f in Q[x] over rational points (1971), when the involved polynomials do not factor through x^n, a Chebyshev polynomial, and an indecomposable degree 4 polynomial.