Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the 'everything for everyone' approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra.

Table of Contents

• Preface
• 1 Elementary Number Theory
• 1.1 Divisibility
• 1.2 Primes and factorization
• 1.3 Congruences
• 1.4 Solving congruences
• 1.5 Theorems of Fermat and Euler
• 1.6 RSA cryptosystem
• 2 Groups
• 2.1 Definition of a group
• 2.2 Examples of groups
• 2.3 Subgroups
• 2.4 Cosets and Lagrange's Theorem
• 3 Rings
• 3.1 Definition of a ring
• 3.2 Subrings and ideals
• 3.3 Ring homomorphisms
• 3.4 Integral domains
• 4 Fields
• 4.1 Definition and basic properties of a field
• 5 Finite Fields
• 5.1 Number of elements in a finite field
• 5.2 How to construct finite fields
• 5.3 Properties of finite fields
• 5.4 Polynomials over finite fields
• 5.5 Permutation polynomials
• 5.6 Applications
• 5.6.1 Orthogonal Latin squares
• 5.6.2 Diffie/Hellman key exchange
• 6 Vector Spaces
• 6.1 Definition and examples
• 6.2 Basic properties of vector spaces
• 6.3 Subspaces
• 7 Polynomials
• 7.1 Basics
• 7.2 Unique factorization
• 7.3 Polynomials over the real and complex numbers
• 7.4 Root formulas
• 8 Linear Codes
• 8.1 Basics
• 8.2 Hamming codes
• 8.3 Encoding
• 8.4 Decoding
• 8.5 Further study
• 8.6 Exercises
• 9 Appendix
• 9.1 Mathematical induction
• 9.2 Well-ordering Principle
• 9.3 Sets
• 9.4 Functions
• 9.5 Permutations
• 9.6 Matrices
• 9.7 Complex numbers
• 10 Hints and Partial Solutions to Selected Exercises
• Bibliography