Completely split absolutely irreducible integer-valued polynomials ov...

Non-unique factorization of elements into irreducibles has been observed in the ring of integer-valued polynomials and its generalizations. It is known that every (multi-)set consisting of integers greater than 1 can be realized as the (multi-)set of lengths of an integer-valued polynomial over a Dedekind domain D with infinitely many maximal ideals whose residue fields are finite. Moreover, under the same assumptions on D, Int(D) is not transfer Krull. The proofs of these two statements are build on the constructions of integer-valued polynomial whose factorization behaviour can be fully controlled. For this, it is crucial to avoid the situation of a factorization in which an irreducible factor occurs more than once. This is because in non-unique factorization domains there is in general no saying how the powers of an irreducible element factor. From a factorization theoretic point of view, one wants therefore identify those elements among the irreducibles whose powers factor uniquely. We call an irreducible element f absolutely irreducible or strong atom if every power f n of f has essentially one factorization, namely f · f · · · f (n times). Irreducible elements which are not absolutely irreducible are known to exist in rings of integer-valued polynomials. In a recent project we were able to characterize the absolutely irreducible polynomials among the completely split polynomials in terms of their root set in Int(R) where R is a discrete valuation domain with finite residue field. This is joint work with S. Frisch and S. Nakato.