# Schaum’s Outline Abstract Algebra Book pdf Download 2022 free

Schaum’s Outline Abstract Algebra Book pdf Download 2022

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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 of 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-Hölder 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.

## BOOK Contents

PART I : SETS AND RELATIONS

Chapter 01 :  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 02 : 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 SYSTEM,

Chapter 03 : 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 ſal, 4.9 Addition and Multiplication on Z, 4.10 Other Properties of Integers, Solved Problems, Supplementary Problems,

Chapter 05 : 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 06 : 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 07 : 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 08 : 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 09 : 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 p2 , 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 × 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

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