On Schur's conjecture

Abstract

AbstractWe study polynomials over an integral domainRwhich, for infinitely many prime idealsP, induce a permutation ofR/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field ofR. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.