《代数数论》课程大纲

1、代数数论课程教学大纲课程基本信息(Course Information )课程代码(Course Code)*学时MA4112/MA417( Credit Hours)*学分32(Credits)2*课程名称(Course Name)（中文）代数数论(英文) Algebraic Number Theory课程性质(Course Type)专业方向选修 A组授课对象（Audie nee）有兴趣了解代数数论初步知识的各专业学生授课语言(Language ofIn strueti on)中文或英文*开课院系（School）数学系先修课程（Prerequisite）抽象代数授课教师（Instrueto

2、r）课程网址(Course Webpage)* 课程简介(Description)代数数论是研究代数数域和代数整数的学科，它的理论建立需要综合运用多种数学分 支的工具，它的成果和思想也为许多数学和应用问题的解决起到关键作用。本课程讲 解经典代数数论的最基本概念和内容，展现19世纪数学家（从高斯到希尔伯特）在代数数论领域的工作，并选择若干专题说明这些初等代数数论想法的威力。本课程 具体教学内容和具体教学方式由任课教师依据学生情况和老师背景具体确定。由于 课程学时所限，我们底下的初略的教学内容描述偏向于讨论初等代数数论中的代数理 论。1. 代数数域和代数整数环（6学时）2. 整数环中素理想分解（1

3、4学时）3. 理想类群和单位群（8学时）4. 代数数论想法的应用（4学时）* 课程简介(Description)In the study of algebraic nu mber theory, the main goal is to un dersta nd algebraic nu mber fields and algebraic in tegers. This line of research is supported by many subfields of mathematics and plays important role in solving problems arisin

4、g from all kinds of fields in mathematics and its applications.This course aims to tell the storyabout classical algebraic number theory as developed by those pioneers in 19th century from Gauss to Hilbert. We also try to give some modern applications to show the power and beauty of the ideas of tho

5、se pion eers. The exact content of the course should be determ ined by the in structor accord ing to his/her taste as well as the backgro und of the stude nts sitti ng in the course. Due to the very limited class hours,the rough outl ine of the course structure below is mainly about the algebraic as

6、pect of eleme ntary algebraic nu mber theory:1. Algebraic nu mber fields and algebraic in tegers (6 hours)2. Factorization into prime ideals (14 hours)3. The ideal class group and the unit group (8 hours)4. Applicati ons (4 hours)课程教学大纲（course syllabus）*学习目标(Learning Outcomes)1代数数域和代数整数环（A4,A5）1.1代数

7、数域1.2代数整数环2整数环中的素理想分解（B1,B2,B3,B72.1分解的存在惟一性2.2分歧指数，剩余类域次数和分裂次数2.3伽罗瓦扩域中的素理想分解2.4 Kronecker-Weber 定理3理想类群和单位群（B1,B2,B3,B73.1类群和类数3.2 Dirichlet单位定理4应用举例（图上伽罗瓦理论，符号动力系统分类中的数论问题，编码理论应用等）（C1,C4*教学内容、进度安排 及要求(Class Schedule& Requireme nts)教学内容学时教学方式作业及要求基本要求考查方式代数数域和代数整数环6面授习题完成要求书面作业 和课堂报告整数环中的 素理想分

8、解14面授习题完成要求书面作业和课堂报告理想类群和单位群8面授习题完成要求书面作业 和课堂报告代数数论想 法的应用4面授习题完成要求书面作业和课堂报告*考核方式(Gradi ng)由任课教师根据实际教学情况掌握*教材或参考资料(Textbooks & OtherMaterials)1. 冯克勤，代数数论，科学出版社，2001.2. 冯克勤，代数数论简史，湖南教育出版社，2002.3. /filaseta/gradcourses/Math784.html4. http:/www.math.uc onn. edu/kc on rad/blurbs/5. I.N. Stewart and D.O. Tall. Algebraic Number Theory. 3rd editio n, A.K. Peters, 2002.6. M.R. Murty, J. Esm on de, Problems in Algebraic Number Theory, 2rd editi on, Spri nger, 2005.7. H.M. Stark, Galois theory, algebraic number theory, and Zeta functions, in: From Number Theory to P