Introduction to Abstract Algebra (eBook)
291 Seiten
Elsevier Science (Verlag)
978-0-08-092488-5 (ISBN)
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.
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.
Front Cover 1
Introduction to Abstract Algebra 4
Copyright Page 5
Table of Contents 6
Dedication 5
PREFACE 8
ACKNOWLEDGMENTS 12
Chapter 0. NUMBERS 14
0.1 A Naïve Survey of Real Numbers 14
0.2 Basic Theorems on Integers: A Heuristic Look 19
0.3 Complex Numbers: Normal Form 25
0.4 Complex Numbers: Polar Form 34
0.5 Complex Numbers: Root Extractions 41
Chapter 1. SETS TO GROUPS 49
1.1 Sets 49
1.2 Induction and Well Ordering 56
1.3 Functions or Mappings 64
1.4 Semigroups 73
1.5 Groups: Number Systems 80
1.6 Groups: Other Examples 87
1.7 Isomorphism 93
Chapter 2. ELEMENTARY THEORY OF GROUPS 100
2.1 Elementary Properties 100
2.2 Subgroups 107
2.3 The Euclidean Group 112
2.4 Cyclic Groups 120
2.5 Permutation Groups 126
2.6 Cycles and the Parity Theorem 135
2.7 Cosets and Lagrange's Theorem 143
Chapter 3. ELEMENTARY THEORY OF RINGS 151
3.1 Definition and Examples 151
3.2 Elementary Properties 158
3.3 Types of Rings I 166
3.4 Types of Rings II 173
3.5 Characteristic and Quaternions 178
Chapter 4. QUOTIENT OR FACTOR SYSTEMS 186
4.1 Equivalence Relations and Partitions 186
4.2 Congruences mod n 192
4.3 Congruence Classes and Zn 200
4.4 Normal Subgroups and Quotient Groups 205
4.5 Ideals and Quotient Rings 212
4.6 Homomorphism 220
Chapter 5. POLYNOMIAL RINGS 231
5.1 The Polynomial Ring R[x] 231
5.2 Division Algorithm in Z and F[x] 238
5.3 Euclidean Algorithm in Z and F[x] 245
5.4 Unique Factorization in Z and F[x] 253
5.5 Zeros of Polynomials 259
5.6 Rational Polynomials 267
5.7 Quotient Polynomial Rings 275
ANSWERS OR HINTS TO SELECTED ODD-NUMBERED PROBLEMS 284
INDEX 300
| Erscheint lt. Verlag | 28.6.2014 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Technik | |
| Wirtschaft | |
| ISBN-10 | 0-08-092488-3 / 0080924883 |
| ISBN-13 | 978-0-08-092488-5 / 9780080924885 |
| Haben Sie eine Frage zum Produkt? |
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