循环矩阵与多项式
发布时间：2018-10-30 13:22
【摘要】：本文主要分为三章,以循环矩阵和K?nig-Rados定理为基础,分别应用在多项式互素,分圆多项式,本原多项式等.第一章为预备知识,主要介绍了有限域中特征,本原元,单位根,循环矩阵,分圆多项式以及本原多项式等的定义以及重要的性质定理.第二章中,以K?nig-Rados定理为基础,将取值范围扩至n次单位根群,得到了K?nig-Rados定理的推广.更进一步的,将推广后的定理分别应用到多项式互素,分圆多项式以及本原多项式的判定上,分别得到了多项式xf)(与-1mx互素的充要条件(所需条件和取值范围关系密切,故可根据取值范围分情况讨论),分圆多项式与本原多项式的判定条件.第三章中,我们对全文进行总结,并在此基础上提出部分有待改进和完善的问题供大家进一步研究。
[Abstract]:Based on the cyclic matrix and K?nig-Rados theorem, this paper is mainly divided into three chapters, which are applied to polynomial coprime, circular polynomial, primitive polynomial and so on. In the first chapter, we introduce the definitions and important theorems of the characteristics, primitive elements, unit roots, cyclic matrices, circular polynomials and primitive polynomials in the finite domain. In chapter 2, based on K?nig-Rados theorem, the range of values is extended to the unit root group of degree n, and the generalization of K?nig-Rados theorem is obtained. Furthermore, the generalized theorem is applied to the determination of polynomial coprime, circular polynomial and primitive polynomial respectively. The sufficient and necessary conditions of polynomial xf) (and 1mx coprime are obtained respectively. Therefore, it can be discussed according to the value range), the judgment conditions of the circular polynomial and the primitive polynomial. In the third chapter, we summarize the full text and put forward some questions for further study.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21;O174.14
[Abstract]:Based on the cyclic matrix and K?nig-Rados theorem, this paper is mainly divided into three chapters, which are applied to polynomial coprime, circular polynomial, primitive polynomial and so on. In the first chapter, we introduce the definitions and important theorems of the characteristics, primitive elements, unit roots, cyclic matrices, circular polynomials and primitive polynomials in the finite domain. In chapter 2, based on K?nig-Rados theorem, the range of values is extended to the unit root group of degree n, and the generalization of K?nig-Rados theorem is obtained. Furthermore, the generalized theorem is applied to the determination of polynomial coprime, circular polynomial and primitive polynomial respectively. The sufficient and necessary conditions of polynomial xf) (and 1mx coprime are obtained respectively. Therefore, it can be discussed according to the value range), the judgment conditions of the circular polynomial and the primitive polynomial. In the third chapter, we summarize the full text and put forward some questions for further study.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21;O174.14
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