On Schur's conjecture

Abstract

Abstract We study polynomials over an integral domain R which, for infinitely many prime ideals P , induce a permutation of R/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 of R . 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.