On Certain Divisibility Property of Polynomials over Integral Domains

Abstract

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