On Certain Divisibility Property of Polynomials over Integral Domains

Luis F. Caceres, Jose A Velez-Marulanda


An integral domain $R$ is a \textit{degree-domain} if for given two polynomials $f(x)$ and $g(x)$ in $R[x]$ such that for all $k\in R$ $(g(k)\not=0\Rightarrow g(k)|f(k))$, then  $f(x)=0$ or $\deg f\geq \deg g$. We prove that the ring of integers $\mathcal{O}_L$ is a degree-domain, where $\mathbb{Q}\subseteq L$ is a finite Galois extension. Then we study degree-domains that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations

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DOI: https://doi.org/10.5539/jmr.v3n3p28

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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