# Introduction to Abstract Algebra

Introduction to Abstract Algebra provides insight into the methods of abstract algebra. This book provides information pertinent to the fundamental concepts of abstract algebra.Organized into five chapters, this book begins with an overview of the study of natural numbers that are used historically for the purpose of counting the objects in different assemblages. This text then examines the concepts of set and elements of a set. Other chapters contain an intuitive survey of the different kinds of real numbers, with the inclusion of many very important results on integers. This book presents as well a brief survey of algebraic systems from the trivial sets to the more highly structures groups, with emphasis on the elementary properties of groups. The final chapter deals with the simple development of complex numbers.This book is intended to be suitable for students in abstract algebra.

Hardbound, 291 Pages

Published: February 1975

ISBN: 978-0-12-505750-9

## Contents

• Preface

Acknowledgments

Chapter 0 Numbers

0.1 A NaÃ¯ve 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