Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Abstract

Let D be a unique factorization domain and S an infinite subset of D. If f ( X ) is an element in the ring of integer-valued polynomials over S with respect to D (denoted Int ( S , D ) ), then we characterize the irreducible elements of Int ( S , D ) in terms of the fixed-divisor of f ( X ) . The characterization allows us to show that every nonzero rational number n / m is the leading coefficient of infinitely many irreducible polynomials in the ring Int ( Z ) = Int ( Z , Z ) . Further use of the characterization leads to an analysis of the particular factorization properties of such integer-valued polynomial rings. In the case where D = Z , we are able to show that every rational number greater than 1 serves as the elasticity of some polynomial in Int ( S , Z ) (i.e., Int ( S , Z ) is fully elastic).