Table of Contents

Preface

Acknowledgments

Chapter 0 Numbers

0.1 A Naïve Survey of Real Numbers

0.2 Basic Theorems on Integers : A Heuristic Look

0.3 Complex Numbers: Normal Form

0.4 Complex Numbers: Polar Form

0.5 Complex Numbers: Root Extractions

Chapter 1 Sets to Groups

1.1 Sets

1.2 Induction and Well Ordering

1.3 Functions or Mappings

1.4 Semigroups

1.5 Groups: Number Systems

1.6 Groups: Other Examples

1.7 Isomorphism

Chapter 2 Elementary Theory of Groups

2.1 Elementary Properties

2.2 Subgroups

2.3 The Euclidean Group

2.4 Cyclic Groups

2.5 Permutation Groups

2.6 Cycles and the Parity Theorem

2.7 Cosets and Lagrange's Theorem

Chapter 3 Elementary Theory of Rings

3.1 Definition and Examples

3.2 Elementary Properties

3.3 Types of Rings I

3.4 Types of Rings II

3.5 Characteristic and Quaternions

Chapter 4 Quotient or Factor Systems

4.1 Equivalence Relations and Partitions

4.2 Congruences Mod n

4.3 Congruence Classes and Zn

4.4 Normal Subgroups and Quotient Groups

4.5 Ideals and Quotient Rings

4.6 Homomorphism

Chapter 5 Polynomial Rings

5.1 The Polynomial Ring R[x]

5.2 Division Algorithm in Z and F[x]

5.3 Euclidean Algorithm in Z and F[x]

5.4 Unique Factorization in Z and F[x]

5.5 Zeros of Polynomials

5.6 Rational Polynomials

5.7 Quotient Polynomial Rings

Answers or Hints to Selected Odd-Numbered Problems

Index