 Prerequisite：Linear Algebra
 Recommended for： junior
 Introduction：
Abstract algebra is the subject area in mathematics that studies algebraic structures of mathematical objects, such as integers, complex numbers, matrices, roots of polynomials, symmetries of polyhedrons, and et al. Contemporary mathematices make extensive use of abstract algebra.
For example, algebraic number theory and algebraic geometry apply algebraic methods to problems in number theory and geometry, respectively. In recent years, abstract algebra also figures prominently in coding theory and cryptography, which are essential in telecommunication nowadays. In this oneyear course, we will study three kinds of algebraic structures that appear most often in mathematics, namely, groups, rings, and fields, culminating in the proof of the theorem that a general quintic equation has no solutions in radicals.
 Groups<ol>Basic properties</ol><ol>Subgroups, cosets and Lagrange's theorem</ol><ol>Normal subgroups and quotient groups</ol><ol>Homomorphism theorems</ol><ol>Cayley's theorem</ol><ol>Direct product</ol><ol>Finite abelian groups</ol><ol>Cauchy's and Sylow's theorems</ol><ol>Optional topics: free groups, series of a group, ??group actions, matrix groups</ol><ol>Polya theorem for counting, coset decoding in coding theory</ol>

Rings<ol>Basic properties</ol><ol>Subrings Ideals and quotient rings</ol><ol>Maximal and prime ideals</ol><ol>Integral domains and fields of quotients</ol><ol>Polynomial rings</ol><ol>Euclidean rings, principal ideal rings, unique factorization domains</ol>

Fields<ol>Basic properties</ol><ol>Subfields and field extensions</ol><ol>Roots of polynomials</ol><ol>Construction with straight edge and compass finite fields</ol><ol>Optional topics: Galois theory, solvability by radicals</ol>
 John B. Fraleigh: A First Course in Abstract Algebra, 5ed., AddisonWesley 1993

Joseph A. Gallian: Contemporary Abstract Algebra, 3ed., D. C. Heath & Co. 1994

I. N. Herstein: Abstract Algebra, 3ed., Prentice Hall 1996

I. N. Herstein: Topics in Algebra, 2ed., John Wiley & Sons 1975
