Irreducible polynomials in Int(Z)

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 divisord. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. Wedescribe a computational method which allows us to recognize all irreducible polynomials in Int(Z). We present some known facts, preliminary new results and open questions.