课程详述

COURSE SPECIFICATION

以下课程信息可能根据实际授课需要或在课程检讨之后产生变动。如对课程有任何疑问，请联

系授课教师。

The course information as follows may be subject to change, either during the session because of unforeseen

circumstances, or following review of the course at the end of the session. Queries about the course should be

directed to the course instructor.

课程名称 Course Title

数论专题 Topics in Number Theory

授课院系

Originating Department

数学系 Department of Mathematics

课程编号

Course Code

MA319

课程学分 Credit Value

课程类别

Course Type

专业选修课 Major Elective Courses

授课学期

Semester

夏季 Summer

授课语言

Teaching Language

中英双语 English & Chinese

授课教师、所属学系、联系方

式（如属团队授课，请列明其

他授课教师）

Instructor(s), Affiliation&

Contact

（For team teaching, please list

all instructors）

吴正尧，（外聘教师）汕头大学数学系

Wu Zhengyao, Department of Mathematics, Shantou University

Email: wuzhengyao@stu.edu.cn

Webpag: http://wuzhengyao.oschina.io/homepage/

实验员/助教、所属学系、联系

方式

Tutor/TA(s), Contact

待公布 To be announced

10.

选课人数限额(可不填)

Maximum Enrolment

（Optional）

授课方式

Delivery Method

讲授

Lectures

习题/辅导/讨论

Tutorials

实验/实习

Lab/Practical

其它(请具体注明)

Other（Please specify）

总学时

Total

11.

学时数

Credit Hours

12.

先修课程、其它学习要求

Pre-requisites or Other

Academic Requirements

初等数论 (MA209-16)、抽象代数 (MA214)或者抽象代数（ H） (MA219)、数学分析

III(MA203a)或数学分析精讲(MA213-16)

Elementary Number Theory (MA209-16), Abstract Algebra(MA214), Mathematical

Analysis III (MA203a)or Real Analysis(MA213-16)

13.

后续课程、其它学习规划

Courses for which this course

is a pre-requisite

代数曲线等更高等的数论或代数几何课程

More advanced courses in number theory or algebraic geometry, such as Algebraic

Curves

14.

其它要求修读本课程的学系

Cross-listing Dept.

教学大纲及教学日历 SYLLABUS

15.

教学目标 Course Objectives

本课程为数学与应用数学专业学生设计，是抽象代数等课程的后续课程。课程旨在引导学生学习现代数论的重要专题知

识，为有志于在数论或代数几何方向深入学习和研究的高年级学生打下扎实的知识基础。作为专题课，本课程每次开课的

主要专题可能随授课教师而稍有变化。主要的可选主题包括：数域、代数整数、离散赋值和离散赋值环、弱逼近定理、完

备离散赋值域及其扩张、局部域、类域论简介等。

This course is a subsequent course to the Abstract Algebra course for students majored in pure and applied

mathematics. It aims at leading students into selected topics in modern number theory, and for those who are interested

in further study and research in number theory or algebraic geometry, the course will help them to lay down a solid

foundation in background knowledge. As a course in selected topics, the contents may vary slightly each year according

to the instructor. The main topics to be covered include：Number fields, algebraic integers, discrete valuations and

discrete valuation rings, weak approximation theorem, complete discrete valuation fields and their extensions, local fields,

introduction to class field theory, etc.

16.

预达学习成果 Learning Outcomes

通过对本课程的学习，学生能够理解和掌握现代数论的若干重要理论，包括代数数域，局部域，分歧理论等。同时，学生

应当逐渐培养出较好的深入自学能力以及独立钻研科研课题的能力。

An adequate training through this course should help the students to understand some important theories in modern

number theory, such as algebraic number fields, local fields and ramification theory. Also, students are expected to

gradually foster the ability of deep self-teaching and independent, innovative study of research topics.

17.

课程内容及教学日历（如授课语言以英文为主，则课程内容介绍可以用英文；如团队教学或模块教学，教学日历须注明

主讲人）

Course Contents (in Parts/Chapters/Sections/Weeks. Please notify name of instructor for course section(s), if

this is a team teaching or module course.)

第一章环论和域论补遗 (8h)

1.1 诺特环

1.2 模论基础

1.3 局部环与局部化

1.4 域扩张及其迹与范数

1.5 Galois 理论

第二章域的绝对值与离散赋值 (14h)

2.1 域的绝对值及其等价类

2.2 离散赋值

2.3 有理数域的绝对值

2.4 弱逼近定理

2.5 赋值域的完备化

2.6 Hensel 引理

2.7 非阿绝对值的扩张

2.8 Newton 折线

第三章局部域及其扩张 (10h)

3.1 局部域的定义和性质

3.2 非分歧扩张

3.3 完全分歧扩张

3.4 分歧子群

3.5 Krasner 引理

3.6 局部类域论简介

Chapter 1 Complements in ring theory and field theory (8h)

1.1 Noetherian rings

1.2 Introduction to module theory

1.3 Local rings and localization

1.4 Field extensions and their traces and norms

1.5 Galois theory

Chapter 2 Absolute values and discrete valuations of fields (14h)

2.1 Absolute values of fields and their equivalence classes

2.2 Discrete valuations

2.3 Absolute values of the field of rational numbers

2.4 Weak approximation theorem

2.5 Completions of valued fields

2.6 Hensel’s lemma

2.7 Extensions of non-archimedean absolute values

2.8 Newton’s polygon

Chapter 3 Local fields an their extensions (10h)

3.1 Definition and properties of local fields

3.2 Unramified extensions

3.3 Totally ramified extensions

3.4 Ramification subgroups

3.5 Krasner’s lemma

3.6 Introduction to local class field theory

18.

教材及其它参考资料 Textbook and Supplementary Readings

教材 Textbook:

J. S. Milne's notes: Algebraic Number Theory, available at https://www.jmilne.org/math/CourseNotes/ant.html

推荐参考书 Supplementary Readings:

J.-P. Serre, Local Fields, Graduate Texts in Math. no.67, Springer, 1979

课程评估 ASSESSMENT

19.

评估形式

Type of

Assessment

评估时间

Time

占考试总成绩百分比

% of final

score

违纪处罚

Penalty

备注

Notes

出勤 Attendance

课堂表现

Class

Performance

小测验

Quiz

课程项目 Projects

平时作业

Assignments

期中考试

Mid-Term Test

期末考试

Final Exam

期末报告

Final

Presentation

其它（可根据需要

改写以上评估方

式）

Others (The

above may be

modified as

necessary)

20.

记分方式 GRADING SYSTEM

 A. 十三级等级制 Letter Grading

 B. 二级记分制（通过/不通过） Pass/Fail Grading

课程审批 REVIEW AND APPROVAL

21.

本课程设置已经过以下责任人/委员会审议通过

This Course has been approved by the following person or committee of authority