课程大纲

Teaching programme of abstract algebras

Instructor: Yanhua Wang

Class hours:Mon: 15:25--17:05, Wed: 15:25--17:05

Office hour:Mon: 10:00--11:00, 13:30--15:00

Wed: 10:00--11:00, 13:30--15:00

Office Address: Room 721，Hongwa Building

E-mail: yhw@mail.shufe.edu.cn

Course Meet: 09/5/2022—1/1/2023

Class Room:1104

Text book: Thomas W. Judson, Abstract algebra:Theory and Applications, Orthogonal Publishing L3C,2017.

Reference:
(1) Joseph A. Gallian: Contemporary abstract algebra, 2010.
(2) Joseph J. Rotman: A First Course in Abstract Algebra, 2006
(3) Derek J. S. Robinson，An Introduction to Abstract Algebra，2003 (4) Menini C, Oystaeyen F.V., Abstract algebra, 2004

Prerequisites: Linear algebra, Advanced algebra.

Course Objectives: Abstract algebra is a common name for the subarea that studies algebraic structures in their own right. Such structures include groups, rings, fields, modules, vector spaces, and algebras. The specific term abstract algebra was coined at the beginning of the 20th century to distinguish this area from the other parts of algebra. The term modern algebra has also been used to denote abstract algebra.

Grading:

There will have one midterm exam, one final exam, a number of homework assignments and randomly in-class quizzes. Grades will be determined as follows:

Quizzes and Homework:  10%

Midterm Exams:             20%

Final Exam :                   70%

Course Contents:

1. Preliminaries:

1.1. Sets
1.2. Maps
1.3. Equivalence Relations
1.4. Equivalence classes
1.5. The Division Algorithm.

Teaching Aim:

Introduction of definitions of set and equivalence. Review some theorem of Sets. How to show a relation is an equivalence relation. To learn The Division Algorithm.

Aim of Ideological and Political Education:

Introduction the Chinese Remainder Theorem. Students review the long history of China. Students have ardent love for the motherland. Mathematics learning adheres to the principle of serving the motherland, innovating in theory, pursuing excellence, solving frontier problems, integrating theory with practice, independent thinking and self-discipline.

To understand math is abstract. Students will feel the beauty of scientific research, will understand the hard to win research results. Students need to have perseverance, tireless and the spirit of assiduous study.

2. Groups

2.1. Definitions and Examples

2.2. Subgroups
2.3. Cyclic Groups
2.4. Permutation Groups

2.5. Cosets and Lagrange's Theorem
2.6. Isomorphism
2.7. Normal subgroups and factor groups
2.8. Homomorphisms and isomorphism theorems

Teaching Aim:

Definition of group and subgroup. Examples of groups and subgroups. The Integers mod n and Symmetries. Learn what is cyclic group, give examples of cyclic group. Theorem of cyclic group. Learn what is permutation group, examples of cyclic group. Theorem of cyclic group. Compute the composition of permutation groups. The definition of cosets, examples of cosets. Understanding the Lagrange's Theorem and applying the theorem. To Learn Fermat's and Euler's Theorem.

Aim of Ideological and Political Education:

Using the definitions of group and field in this chapter, every student understand he is not lonely. They should help each other, then they can make progress together.

Students understand the idea from concrete to abstract, understand the preciseness and interest of science, and set the goal of exploring the world and science. Through training students to learn how to establish contact between two objects, let students understand that many things are interrelated, so as to establish correct values and outlook on life.

3. Rings and fields

3.1. Rings, Integral Domains and Fields

3.2. Ring Homomorphisms and Ideals

3.3. Maximal Ideals and Prime Ideals

3.4. Application of Codes

Teaching Aim:

Rings, Integral Domains and Fields, Definitions and examples of fields, extension field, splitting field. Ring ideal and Homomorphisms, Maximal and Prime Ideals. The definition of Cryptography, Private Key Cryptography and Public Key Cryptography. To learn and understand all kinds of codes. The theory of codes.

Aim of Ideological and Political Education:

Using the theorem of code in this chapter, science is very important. Students should study hard and work hard. Everyone gives his modest contribution to the strength of the motherland. Through the understanding of mathematical concepts and theories, we can understand that there are many things in nature to see the essence through the phenomenon and grasp the core problems.

In order to solve scientific problems, promote social progress and benefit for people, we should cultivate students' imagination and creativity in future study and work, and realize natural abstract reasoning from concrete to general.

 

Course Schedule (subject to change)

Weeks	Contents
1	Sets, Maps
2	Equivalence Relations and Equivalence Classes
3	The Division Algorithm
4	Group, Subgroup, Cyclic Groups and Permutation Groups
5	Cosets and Lagrange's Theorem
6	Isomorphisms of Group and Normal Subgroups
7	Factor Groups; Midterm Exams
8	Homomorphisms and Isomorphism Theorems
9	Rings, Integral Domains and Fields
10	Ring Homomorphisms and Ideals
11	Maximal Ideal and Prime Ideals
12	Review