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Full Description
A reader-friendly introduction to modern algebra with
important examples from various areas of mathematics
Featuring a clear and concise approach, An Introduction to
Essential Algebraic Structures presents an integrated approach
to basic concepts of modern algebra and highlights topics that play
a central role in various branches of mathematics. The authors
discuss key topics of abstract and modern algebra including sets,
number systems, groups, rings, and fields. The book begins with an
exposition of the elements of set theory and moves on to cover the
main ideas and branches of abstract algebra. In addition, the book
includes:
Numerous examples throughout to deepen readers? knowledge
of the presented material
An exercise set after each chapter section in an effort to
build a deeper understanding of the subject and improve knowledge
retention
Hints and answers to select exercises at the end of the
book
A supplementary website with an Instructors Solutions
manual
An Introduction to Essential Algebraic Structures is
an excellent textbook for introductory courses in abstract algebra
as well as an ideal reference for anyone who would like to be more
familiar with the basic topics of abstract algebra.
Contents
Preface vii
1 Sets 1
1.1 Operations on Sets, 1
Exercise Set 1.1, 7
1.2 Set Mappings, 9
Exercise Set 1.2, 15
1.3 Products of Mappings and Permutations, 16
Exercise Set 1.3, 26
1.4 Operations on Matrices, 28
Exercise Set 1.4, 35
1.5 Binary Algebraic Operations and Equivalence Relations, 37
Exercise Set 1.5, 47
2 Numbers 51
2.1 Some Properties of Integers: Mathematical Induction, 51
Exercise Set 2.1, 55
2.2 Divisibility, 56
Exercise Set 2.2, 63
2.3 Prime Factorization: The Fundamental Theorem of Arithmetic, 64
Exercise Set 2.3, 67
2.4 Rational Numbers, Irrational Numbers, and Real Numbers, 68
Exercise Set 2.4, 76
3 Groups 79
3.1 Groups and Subgroups, 79
Exercise Set 3.1, 93
3.2 Cosets and Normal Subgroups, 94
Exercise Set 3.2, 106
3.3 Factor Groups and Homomorphisms, 108
Exercise Set 3.3, 116
4 Rings 119
4.1 Rings, Subrings, Associative Rings, 119
Exercise Set 4.1, 131
4.2 Rings of Polynomials, 133
Exercise Set 4.2, 142
4.3 Ideals and Quotient Rings, 143
Exercise Set 4.3, 153
4.4 Homomorphisms of Rings, 155
Exercise Set 4.4, 165
5 Fields 169
5.1 Fields: Basic Properties and Examples, 169
Exercise Set 5.1, 180
5.2 Some Field Extensions, 182
Exercise Set 5.2, 187
5.3 Fields of Algebraic Numbers, 187
Exercise Set 5.3, 196
Hints and Answers to Selected Exercises 199
Chapter 1, 199
Chapter 2, 205
Chapter 3, 210
Chapter 4, 214
Chapter 5, 222
Index 225
