A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator. Issue 9 (1st September 2020)

• Record Type: Journal Article
• Authors: Frisch, Sophie; Nakato, Sarah
• Is Part Of: Communications in algebra. Volume 48:Issue 9(2020)
• Page Start: 3716
• Page End: 3723
• Publication Date: 2020-09-01
• Subjects: Factorization -- non-unique factorization -- irreducible elements -- absolutely irreducible elements -- atoms -- strong atoms -- atomic domains -- integer-valued polynomials -- simple graphs -- connected graphs; 13A05 -- 13B25 -- 13F20 -- 11R09 -- 11C08 -- 13P05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1744618
• Languages: English
• ISSNs: 0092-7872

Abstract

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 Int ( D ) = { f ∈ K [ x ] | f ( D ) ⊆ D }, 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.