Course Introduction

• 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 one-year 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.

• Syllabus：

1. Groups
   Basic properties
   Subgroups, cosets and Lagrange's theorem
   Normal subgroups and quotient groups
   Homomorphism theorems
   Cayley's theorem
   Direct product
   Finite abelian groups
   Cauchy's and Sylow's theorems
   Optional topics: free groups, series of a group, ??group actions, matrix groups
   Polya theorem for counting, coset decoding in coding theory
2. Rings
   Basic properties
   Subrings Ideals and quotient rings
   Maximal and prime ideals
   Integral domains and fields of quotients
   Polynomial rings
   Euclidean rings, principal ideal rings, unique factorization domains
3. Fields
   Basic properties
   Subfields and field extensions
   Roots of polynomials
   Construction with straight edge and compass finite fields
   Optional topics: Galois theory, solvability by radicals

• Reference：

1. John B. Fraleigh: A First Course in Abstract Algebra, 5ed., Addison-Wesley 1993
2. Joseph A. Gallian: Contemporary Abstract Algebra, 3ed., D. C. Heath & Co. 1994
3. I. N. Herstein: Abstract Algebra, 3ed., Prentice Hall 1996
4. I. N. Herstein: Topics in Algebra, 2ed., John Wiley & Sons 1975