Irreducibility of integer-valued polynomials I. Issue 3 (11th December 2020)

• Record Type: Journal Article
• Authors: Prasad, Devendra
• Is Part Of: Communications in algebra. Volume 49:Issue 3(2021)
• Page Start: 948
• Page End: 955
• Publication Date: 2020-12-11
• Subjects: Integer-valued polynomials -- irreducible elements; 13F20; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1823990
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let S ⊂ R be an arbitrary subset of a unique factorization domain R and K be the field of fractions of R . The ring of integer-valued polynomials over S is the set Int ( S, R ) = { f ∈ K [ x ] : f ( a ) ∈ R ∀ a ∈ S } . This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call d -sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in Int ( S, R ) . In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in Int ( S, R ) . At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.