# Abstract Algebra

An introduction to the principles and concepts of modern Abstract Algebra. Topics include groups, rings, and fields, isomorphisms, and homomorphisms with applications to number theory, the theory of equations, and geometry.
Prerequisite: Methods of Proof and Linear Algebra with a grade of C or better.

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor. An official syllabus will be provided to the student on the first day of class by the instructor.

## Chapter 1

Groups and Subgroups

Lessons Homework
1.1 Introduction and Examples 1.1
1.2 Binary Operations 1.2
1.3 Isomorphic Binary Structures 1.3
1.4 Groups 1.4
1.5 Subgroups 1.5
1.6 Cyclic Groups 1.6
1.7 Generators and Cayley Diagraphs 1.7

## Chapter 2

Permutations, Cosets, and Direct Products

Lessons Homework
2.1 Groups of Permutations 2.1
2.2 Orbits, Cycles, and the Alternating Group 2.2
2.3 Cosets and the Theorem of Lagrange 2.3
2.4 Direct Products and Finitely Generated Abelian Groups 2.4

## Chapter 3

Homomorphisms and Factor Groups

Lessons Homework
3.1 Homomorphisms 3.1
3.2 Factor Groups 3.2
3.3 Factor Group Computations and Simple Groups 3.3
3.4 Group action on a set 3.4

## Chapter 4

Rings and Fields

Lessons Homework
4.1 Rings and Fields 4.1
4.2 Integral Domains 4.2
4.3 Fermat's and Euler's Theorem 4.3
4.4 The Field of Quotients of an Integral Domain 4.4
4.5 Ring of Polynomials 4.5
4.6 Factorization of Polynomials Over a Field 4.6
4.7 Noncommutative Examples 4.7

## Chapter 5

Ideals and Factor Rings

Lessons Homework
5.1 Homomorphisms and Factor Rings 5.1
5.2 Prime and Maximal Ideals 5.2