On Certain Divisibility Property of Polynomials
Luis F. Caceres, Jose A Velez Marulanda
TL;DR
This paper explores the properties of D-rings, providing new characterizations and elementary proofs, especially for rings of algebraic integers and UFDs, to understand polynomial divisibility better.
Contribution
It offers an alternative characterization of D-rings and an elementary proof that rings of algebraic integers are D-rings, enhancing understanding of polynomial divisibility.
Findings
Rings of algebraic integers are D-rings.
Characterization of D-rings that are UFDs for polynomial divisibility.
Elementary proof techniques for D-ring properties.
Abstract
We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Polynomial and algebraic computation