【摘要】：本文主要分為三章,以循環(huán)矩陣和K?nig-Rados定理為基礎(chǔ),分別應(yīng)用在多項(xiàng)式互素,分圓多項(xiàng)式,本原多項(xiàng)式等.第一章為預(yù)備知識(shí),主要介紹了有限域中特征,本原元,單位根,循環(huán)矩陣,分圓多項(xiàng)式以及本原多項(xiàng)式等的定義以及重要的性質(zhì)定理.第二章中,以K?nig-Rados定理為基礎(chǔ),將取值范圍擴(kuò)至n次單位根群,得到了K?nig-Rados定理的推廣.更進(jìn)一步的,將推廣后的定理分別應(yīng)用到多項(xiàng)式互素,分圓多項(xiàng)式以及本原多項(xiàng)式的判定上,分別得到了多項(xiàng)式xf)(與-1mx互素的充要條件(所需條件和取值范圍關(guān)系密切,故可根據(jù)取值范圍分情況討論),分圓多項(xiàng)式與本原多項(xiàng)式的判定條件.第三章中,我們對(duì)全文進(jìn)行總結(jié),并在此基礎(chǔ)上提出部分有待改進(jìn)和完善的問題供大家進(jìn)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.
【學(xué)位授予單位】：寧波大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O151.21;O174.14

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本文編號(hào)：2300119