Irreducible polynomials in Int(ℤ)

¹ Ohio State University, 231 W. 18th Ave., MW 505, Columbus, OH 43210, USA
² Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria
³ Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

^* antoniou.6@buckeyemail.osu.edu
^** snakato@tugraz.at
^*** roswitha.rissner@aau.at

Abstract

In order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g/d] is irreducible in the case where d is a square-free integer and g ∈ ℤ[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreduciblexc polynomials in Int(ℤ). We present some known facts, preliminary new results and open questions.