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

In rings which allow non-unique factorizations it is possible that powers of an irreducible element f have a factorization different from the trivial one f^n = f · f · · · f; those irreducibles all of whose powers factor uniquely are called absolutely irreducible. It is in general hard to identify the absolutely irreducibles explicitly among the irreducibles. However, in order to understand the factorization behaviour in non-unique factorization domains (or more generally monoids) it is crucial to get a handle on powers of irreducibles. From a factorization point of view, rings of integer-valued polynomials provide a rich source of examples of interesting behavior. 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 from Graz University of Technology, Austria.