Polynomials with Equal Images over Number Fields

Abstract

Chapman and Ponomarenko [1] characterized when two polynomials f, g ∈ Q[x] have thesame image f(Z) = f(Z). We extend this result to rings of integers in number fields. In particular, if K is a finite extension of Q and O is the ring of algebraic integers in K, we characterize when polynomials f, g ∈ K[x] satisfy f(O) = g(O). As part of our proof, we give a variant of Hilbert’s irreducibility theorem.