中文摘要：

研究了任意域上多项式f（x）在m（≥degf（x））次单位根群中的零点个数与由f（x）的系数所构成的循环矩阵的秩之间的关系,推广了Konig-Rados定理,得到了f（x）与x^m-1互素的充要条件,并给出了分解f（x）及判定f（x）是否为分圆多项式的方法.

英文摘要：

We show that the number of zeros of the polynomial f（x）in the group of m-th roots of unity can be expressed in terms of the rank of the associated left circulant matrix,which generalizes the Konig-Rado theorem.As an application,the sufficient and necessary condition for f（x）to be coprime with x^m-1 is obtained.We also provide the alternative approaches to factorizing f（x）as well as to determining whether f（x）is a cyclotomic polynomial.