Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) by FRANK AYRES and LLOYD R. JAISINGH
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Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) written by
FRANK AYRES , Jr., Ph.D. and
LLOYD R. JAISINGH, Professor of Mathematics, Morehead State University
This book on algebraic systems is designed to be used either as a supplement to current texts or as a stand-alone text for a course in modern abstract algebra at the junior and/or senior levels. In addition, graduate students can use this book as a source for review. As such, this book is intended to provide a solid foundation for future study of a variety of systems rather than to be a study in depth of any one or more. The basic ingredients of algebraic systems–sets of elements, relations, operations, and mappings–are discussed in the first two chapters. The format established for this book is as follows:
. a simple and concise presentation of each topic
. a wide variety of familiar examples
. proofs of most theorems included among the solved problems
. a carefully selected set of supplementary exercises
In this upgrade, the text has made an effort to use standard notations for the set of natural numbers, the set of integers, the set of rational numbers, and the set of real numbers. In addition, definitions are highlighted rather than being embedded in the prose of the text. Also, a new chapter (Chapter 10) has been added to the text. It gives a very brief discussion of Sylow Theorems and the Galois group. The text starts with the Peano postulates for the natural numbers in Chapter 3, with the various number systems of elementary algebra being constructed and their salient properties discussed. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides the reader with much needed practice for the reasoning behind the properties of the abstract systems which follow. The first abstract algebraic system – the Group – is considered in Chapter 9. Cosets of a subgroup, invariant subgroups, and their quotient groups are investigated as well. Chapter 9 ends with the Jordan–Ho¨ lder Theorem for finite groups. Rings, Integral Domains Division Rings, Fields are discussed in Chapters 11–12 while Polynomials over rings and fields are then considered in Chapter 13. Throughout these chapters, considerable attention is given to finite rings. Vector spaces are introduced in Chapter 14. The algebra of linear transformations on a vector space of finite dimension leads naturally to the algebra of matrices (Chapter 15). Matrices are then used to solve systems of linear equations and, thus provide simpler solutions to a number of problems connected to vector spaces. Matrix polynomials are discussed in Chapter 16 as an example of a non-commutative polynomial ring. The characteristic polynomial of a square matrix over a field is then defined. The characteristic roots and associated invariant vectors of real symmetric matrices are used to reduce the equations of conics and quadric surfaces to standard form. Linear algebras are formally defined in Chapter 17 and other examples briefly considered. In the final chapter (Chapter 18), Boolean algebras are introduced and important applications to simple electric circuits are discussed. The co-author wishes to thank the staff of the Schaum’s Outlines group, especially Barbara Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author wishes to thank the estate of Dr. Frank Ayres, Jr. for allowing me to help upgrade the original text.
Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) written by
FRANK AYRES , Jr., Ph.D. and
LLOYD R. JAISINGH
cover the following topics.
PART I SETS AND RELATIONS
1. Sets
Introduction
1.1 Sets
1.2 Equal Sets
1.3 Subsets of a Set
1.4 Universal Sets
1.5 Intersection and Union of Sets
1.6 Venn Diagrams
1.7 Operations with Sets
1.8 The Product Set
1.9 Mappings
1.10 One-to-One Mappings
1.11 One-to-One Mapping of a Set onto Itself
Solved Problems
Supplementary Problems
2. Relations and Operations
Introduction
2.1 Relations
2.2 Properties of Binary Relations
2.3 Equivalence Relations
2.4 Equivalence Sets
2.5 Ordering in Sets
2.6 Operations
2.7 Types of Binary Operations
2.8 Well-Defined Operations
2.9 Isomorphisms
2.10 Permutations
2.11 Transpositions
2.12 Algebraic Systems
Solved Problems
Supplementary Problems
PART II NUMBER SYSTEMS
3. The Natural Numbers
Introduction
3.1 The Peano Postulates
3.2 Addition on N
3.3 Multiplication on N
3.4 Mathematical Induction
3.5 The Order Relations
3.6 Multiples and Powers
3.7 Isomorphic Sets
Solved Problems
Supplementary Problems
4. The Integers
Introduction
4.1 Binary Relation
4.2 Addition and Multiplication on J
4.3 The Positive Integers
4.4 Zero and Negative Integers
4.5 The Integers
4.6 Order Relations
4.7 Subtraction ‘‘’’
4.8 Absolute Value jaj
4.9 Addition and Multiplication on Z
4.10 Other Properties of Integers
Solved Problems
Supplementary Problems
5. Some Properties of Integers
Introduction
5.1 Divisors
5.2 Primes
5.3 Greatest Common Divisor
5.4 Relatively Prime Integers
5.5 Prime Factors
5.6 Congruences
5.7 The Algebra of Residue Classes
5.8 Linear Congruences
5.9 Positional Notation for Integers
Solved Problems
Supplementary Problems
6. The Rational Numbers
Introduction
6.1 The Rational Numbers
6.2 Addition and Multiplication
6.3 Subtraction and Division
6.4 Replacement
6.5 Order Relations
6.6 Reduction to Lowest Terms
6.7 Decimal Representation
Solved Problems
Supplementary Problems
7. The Real Numbers
Introduction
7.1 Dedekind Cuts
7.2 Positive Cuts
7.3 Multiplicative Inverses
7.4 Additive Inverses
7.5 Multiplication on K
7.6 Subtraction and Division
7.7 Order Relations
7.8 Properties of the Real Numbers
Solved Problems
Supplementary Problems
8. The Complex Numbers
Introduction
8.1 Addition and Multiplication on C
8.2 Properties of Complex Numbers
8.3 Subtraction and Division on C
8.4 Trigonometric Representation
8.5 Roots
8.6 Primitive Roots of Unity
Solved Problems
Supplementary Problems
9. Groups
Introduction
9.1 Groups
9.2 Simple Properties of Groups
9.3 Subgroups
9.4 Cyclic Groups
9.5 Permutation Groups
9.6 Homomorphisms
9.7 Isomorphisms
9.8 Cosets
9.9 Invariant Subgroups
9.10 Quotient Groups
9.11 Product of Subgroups
9.12 Composition Series
Solved Problems
Supplementary Problems
10. Further Topics on Group Theory
Introduction
10.1 Cauchy’s Theorem for Groups
10.2 Groups of Order 2p and p2
10.3 The Sylow Theorems
10.4 Galois Group
Solved Problems
Supplementary Problems
11. Rings
Introduction
11.1 Rings
11.2 Properties of Rings
11.3 Subrings
11.4 Types of Rings
11.5 Characteristic
11.6 Divisors of Zero
11.7 Homomorphisms and Isomorphisms
11.8 Ideals
11.9 Principal Ideals
11.10 Prime and Maximal Ideals
11.11 Quotient Rings
11.12 Euclidean Rings
Solved Problems
Supplementary Problems
12. Integral Domains, Division Rings, Fields
Introduction
12.1 Integral Domains
12.2 Unit, Associate, Divisor
12.3 Subdomains
12.4 Ordered Integral Domains
12.5 Division Algorithm
12.6 Unique Factorization
12.7 Division Rings
12.8 Fields
Solved Problems
Supplementary Problems
13. Polynomials
Introduction
13.1 Polynomial Forms
13.2 Monic Polynomials
13.3 Division
13.4 Commutative Polynomial Rings with Unity
13.5 Substitution Process
13.6 The Polynomial Domain F½x
13.7 Prime Polynomials
13.8 The Polynomial Domain C½x
13.9 Greatest Common Divisor
13.10 Properties of the Polynomial Domain F½x
Solved Problems
Supplementary Problems
14. Vector Spaces
Introduction
14.1 Vector Spaces
14.2 Subspace of a Vector Space
14.3 Linear Dependence
14.4 Bases of a Vector Space
14.5 Subspaces of a Vector Space
14.6 Vector Spaces Over R
14.7 Linear Transformations
14.8 The Algebra of Linear Transformations
Solved Problems
Supplementary Problems
15. Matrices
Introduction
15.1 Matrices
15.2 Square Matrices
15.3 Total Matrix Algebra
15.4 A Matrix of Order m n
15.5 Solutions of a System of Linear Equations
15.6 Elementary Transformations on a Matrix
15.7 Upper Triangular, Lower Triangular, and Diagonal Matrices
15.8 A Canonical Form
15.9 Elementary Column Transformations
15.10 Elementary Matrices
15.11 Inverses of Elementary Matrices
15.12 The Inverse of a Non-Singular Matrix
15.13 Minimum Polynomial of a Square Matrix
15.14 Systems of Linear Equations
15.15 Systems of Non-Homogeneous Linear Equations
15.16 Systems of Homogeneous Linear Equations
15.17 Determinant of a Square Matrix
15.18 Properties of Determinants
15.19 Evaluation of Determinants
Solved Problems
Supplementary Problems
16. Matrix Polynomials
Introduction
16.1 Matrices with Polynomial Elements
16.2 Elementary Transformations
16.3 Normal Form of a -Matrix
16.4 Polynomials with Matrix Coefficients
16.5 Division Algorithm
16.6 The Characteristic Roots and Vectors of a Matrix
16.7 Similar Matrices
16.8 Real Symmetric Matrices
16.9 Orthogonal Matrices
16.10 Conics and Quadric Surfaces
Solved Problems
Supplementary Problems
17. Linear Algebras
Introduction
17.1 Linear Algebra
17.2 An Isomorphism
Solved Problems
Supplementary Problems
18. Boolean Algebras
Introduction
18.1 Boolean Algebra
18.2 Boolean Functions
18.3 Normal Forms
18.4 Changing the Form of a Boolean Function
18.5 Order Relation in a Boolean Algebra
18.6 Algebra of Electrical Networks
18.7 Simplification of Networks
Solved Problems
Supplementary Problems
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