Course Introduction

• Prerequisite：Undergraduate Algebra
• Recommended for： graduate students
• Introduction：

Abstract Algebra, sometimes also called modern algebra or algebra in short, is the branch of mathematics concerning the study of algebraic structures such as groups, rings, modules and fields. This course is an advanced study of the above structures.

• Syllabus：

1. Groups：Homomorphisms, Factor Groups , Group, Action on a Set, Isomorphism Theorems, Series of Groups, Sylow’s Theorems.
2. Rings：Factorization of Polynomials over a Field, Homomorphisms and Factor Rings, Prime and Maximal Ideals, Unique Factorization Domains, Euclidean Domains, Gaussian Integers.
3. Modules：Direct Products and Sums of Modules, Free Modules, Vector Spaces, Dual Space and Dual Module, Modules over Principal Rings.
4. Fields： Extension Fields and Vector Spaces, Algebraic Extensions, Roots of Polynomials, Finite Fields, Cyclotomic Polynomials, Automorphisms and Galois Theory.

• Reference：

1. John B. Fraleigh, A First Course in Abstract Algebra, Pearson Education, 2003.
2. I. N. Herstein, Abstract Algebra, Prentice-Hall, 1996.
3. Serge Lang, Algebra , Graduate Texts in Mathematics 211, Springer, 2002. (Sections 1.1-1.8, 2.1-2.5, 3.1-3.7, 5.1-5.5, 6.1, 6.2).