This book introduces the basic notions of abstract algebra to sophomores and perhaps even junior mathematics majors who have a relatively weak background with conceptual courses. It introduces the material with many concrete examples and establishes a firm foundation for introducing more abstract mathematical notions.

Gary Mullen is Professor of Mathematics, The Pennsylvania State University, where he earned his Ph.D. His main interest is finite fields, and is founder of the journal "Finite Fields and Their Introduction." He is also the Editor of The Handbook of Finite Fields published by CRC Press.

James Sellers is Professor and Associate Head for Undergraduate Mathematics, The Pennsylvania State University, where he also earned his Ph.D. He has published many research articles and won awards related to his efforts to advance mathematics education.

Elementary Number Theory

Divisibility

Primes and factorization

Congruences

Solving congruences

Theorems of Fermat and Euler

RSA cryptosystem

Groups

De nition of a group

Examples of groups

Subgroups

Cosets and Lagrange's Theorem

Rings

Defiition of a ring

Subrings and ideals

Ring homomorphisms

Integral domains

Fields

Definition and basic properties of a field

Finite Fields

Number of elements in a finite field

How to construct finite fields

Properties of finite fields

Polynomials over finite fields

Permutation polynomials

Applications

Orthogonal latin squares

Di¿e/Hellman key exchange

Vector Spaces

Definition and examples

Basic properties of vector spaces

Subspaces

Polynomials

Basics

Unique factorization

Polynomials over the real and complex numbers

Root formulas

Linear Codes

Basics

Hamming codes

Encoding

Decoding

Further study

Exercises

Appendix

Mathematical induction

Well-ordering Principle

Sets

Functions

Permutations

Matrices

Complex numbers

Hints and Partial Solutions to Selected Exercises