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摘要摘要本文前面部分介绍了有限域理论的基础知识。然后根据有限域的相关知识，对王鑫和王新梅在[1]中提出的判定不可约多项式及本原多项式的一种高效算法进行了论证。该算法提出了三个条件作为判定有限域上多项式的不可约性的充要条件，并在有限域上多项式不可约的前提下，附加了一个条件作为判定有限域上多项式为本原多项式的充要条件。文章的后面部分，使用MicrosoftVisualStudio2008软件，用c++语言编程实现了有限域上的模运算、乘法运算、快速指数算法、欧几里得算法、整数分解算法等核心模块，并最终实现了王鑫和王新梅在[1]中提出的判定方法，实现了对有限域上的多项式是否为不可约多项式及本原多项式的判定。关键词：有限域不可约多项式本原多项式ABSTRACTWeintroducethebasicknowledgeoffinitefieldstheoryinthefrontofthispaper.Accordingtotheknowledgeoffinitefields,wediscussanefficientalgorithm,whichisusedtodeterminewhetherapolynomialoverfinitefieldsisirreducible(primitive)ornot,proposedbyWangXinandWangXinMeiin[1].Threeconditionsareproposedbyitasanecessaryandsufficientconditiontodetermineirreduciblepolynomialsoverthefinitefield.Andunderthepreconditionthatthepolynomialisirreducibleoverfinitefields,thealgorithmproposesaconditionasthenecessaryandsufficientconditiontodeterminewhetherapolynomialisprimitiveornotoverfinitefields.Inthelatterpart,byusingMicrosoftVisualStudio2008software,wemakethemodeoperations,multiplicationoperations,fastexponentialalgorithm,Euclidalgorithm,integerfactorizationalgorithmmodulescometrueinc++language.AndfinallyachievedthedecisionmethodproposedbyWangXinandWangXinMeiin[1],realizedthedeterminationthatwhetherthepolynomialoverfinitefieldsisirreducible(primitive)ornot.Keywords：finitefieldirreduciblepolynomialsprimitivepolynomials目录i目录第一章绪论......................................................................................................................................11.1研究背景和研究意义........................................................................................................11.2相关领域的研究进展........................................................................................................21.3本文主要的研究成果和内容安排....................................................................................3第二章有限域的基础知识..............................................................................................................52.1群、环、域........................................................................................................................52.2多项式环............................................................................................................................62.3域的有限扩张..................................................................................................................102.4有限域的性质..................................................................................................................122.5有限域上的多项式..........................................................................................................132.6本章小结..........................................................................................................................14第三章有限域上不可约多项式和本原多项式的判定方法..........................................................153.1引言..................................................................................................................................153.2有限域上多项式不可约性的判定..................................................................................153.3有限域上多项式本原性的判定......................................................................................203.4本章小结..........................................................................................................................21第四章程序实现............................................................................................................................234.1程序总流程......................................................................................................................234.2数据结构..........................................................................................................................244.2.1有限域上多项式的表示......................................................................................244.2.2多项式读入模块..................................................................................................244.2.3运算符..................................................................................................................254.3基础算法..........................................................................................................................254.3.1快速指数算法......................................................................................................254.3.2整数分解算法......................................................................................................264.3.3欧几里得算法......................................................................................................274.4不可约性判定..................................................................................................................274.5本原性判定......................................................................................................................28第五章程序测试............................................................................................................................29第六章总结与展望........................................................................................................................31致谢...................