# A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator

Publikation: Beitrag in einer FachzeitschriftArtikelForschung

### Abstract

An irreducible element of a commutative ring is absolutely irreducible
if no power of it has more than one (essentially different) factorization
into irreducibles. In the case of the ring
$\IntD=\{f\in K[x]\mid f(D)\subseteq \IntD\}$, of integer-valued polynomials
on a principal ideal domain $D$ with quotient field $K$,
we give an easy to verify graph-theoretic sufficient condition for an
element to be absolutely irreducible and show a partial converse:
the condition is necessary and sufficient
for polynomials with square-free denominator.
Originalsprache englisch arXiv.org e-Print archive Eingereicht - 22 Dez 2019

### ASJC Scopus subject areas

• !!Algebra and Number Theory

### Fields of Expertise

• Information, Communication & Computing