Schaum's Outline of Abstract Algebra | Schaum's Outline

Schaum's Outline of Abstract Algebra

Authors: Lloyd Jaisingh and Frank Ayres

Published: December 2003

eISBN: 9780071430982 0071430989 | ISBN: 9780071403276

Contents

Part I: Sets and Relations

Chapter 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

Chapter 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

Chapter 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

Chapter 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 lal

4.9 Addition and Multiplication on Z

4.10 Other Properties of Integers

Solved Problems

Supplementary Problems

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Part III: Groups, Rings and Fields

Chapter 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

Chapter 10 Further Topics on Group Theory

Introduction

10.1 Cauchy’s Theorem for Groups

10.2 Groups of Order 2p and p[sup(2)]

10.3 The Sylow Theorems

10.4 Galois Group

Solved Problems

Supplementary Problems

Chapter 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

Chapter 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

Chapter 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 F[x]

13.7 Prime Polynomials

13.8 The Polynomial Domain C[x]

13.9 Greatest Common Divisor

13.10 Properties of the Polynomial Domain F[x]

Solved Problems

Supplementary Problems

Chapter 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

Chapter 15 Matrices

Introduction

15.1 Matrices

15.2 Square Matrices

15.3 Total Matrix Algebra

15.4 A Matrix of Order m x 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

Chapter 16 Matrix Polynomials

Introduction

16.1 Matrices with Polynomial Elements

16.2 Elementary Transformations

16.3 Normal Form of a λ-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

Chapter 17 Linear Algebras

Introduction

17.1 Linear Algebra

17.2 An Isomorphism

Solved Problems

Supplementary Problems

Chapter 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

Index