Softcover ISBN:  9781470474768 
Product Code:  CHEL/330.S 
List Price:  $69.00 
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AMS Member Price:  $62.10 
eBook ISBN:  9781470476038 
Product Code:  CHEL/330.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9781470474768 
eBook: ISBN:  9781470476038 
Product Code:  CHEL/330.S.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 
Softcover ISBN:  9781470474768 
Product Code:  CHEL/330.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470476038 
Product Code:  CHEL/330.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9781470474768 
eBook ISBN:  9781470476038 
Product Code:  CHEL/330.S.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 

Book DetailsAMS Chelsea PublishingVolume: 330; 1988; 626 ppMSC: Primary 00
This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach—emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s—was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
ReadershipUndergraduates and graduate students interested in algebra.

Table of Contents

Front Cover

Preface to the Third Edition

From the Preface to the First Edition

From the Preface to the Second Edition

Contents

List of Symbols

CHAPTER I Sets, Functions, and Integers

1. Sets

2. Functions

3. Relations and Binary Operations

4. The Natural Numbers

5. Addition and Multiplication

6. Inequalities

7. The Integers

8. The Integers Modulo n

9. Equivalence Relations and Quotient Sets

10. Morphisms

11. Semigroups and Monoids

CHAPTER II Groups

1. Groups and Symmetry

2. Rules of Calculation

3. Cyclic Groups

4. Subgroups

5. Defining Relations

6. Symmetric and Alternating Groups

7. Transformation Groups

8. Cosets

9. Kernel and Image

10. Quotient Groups

CHAPTER III Rings

1. Axioms for Rings

2. Constructions for Rings

3. Quotient Rings

4. Integral Domains and Fields

5. The Field of Quotients

6. Polynomials

7. Polynomials as Functions

8. The Division Algorithm

9. Principal Ideal Domains

10. Unique Factorization

11. Prime Fields

12. The Euclidean Algorithm

13. Commutative Quotient Rings

CHAPTER IV Universal Constructions

1. Examples of Universals

2. Functors

3. Universal Elements

4. Polynomials in Several Variables

5. Categories

6. Posets and Lattices

7. Contravariance and Duality

8. The Category of Sets

9. The Category of Finite Sets

CHAPTER V Modules

1. Sample Modules

2. Linear Transformations

3. Submodules

4. Quotient Modules

5. Free Modules

6. Biproducts

7. Dual Modules

CHAPTER VI Vector Spaces

1. Bases and Coordinates

2. Dimension

3. Constructions for Bases

4. Dually Paired Vector Spaces

5. Elementary Operations

6. Systems of Linear Equations

CHAPTER VII Matrices

1. Matrices and Free Modules

2. Matrices and Biproducts

3. The Matrix of a Map

4. The Matrix of a Composite

5. Ranks of Matrices

6. Invertible Matrices

7. Change of Bases

8. Eigenvectors and Eigenvalues

CHAPTER VIII Special Fields

1. Ordered Domains

2. The Ordered Field Q

3. Polynomial Equations

4. Convergence in Ordered Fields

5. The Real Field R

6. Polynomials over R

7. The Complex Plane

8. The Quaternions

9. Extended Formal Power Series

10. Valuations and padic Numbers

CHAPTER IX Determinants and Tensor Products

1. Multilinear and Alternating Functions

2. Determinants of Matrices

3. Cofactors and Cramer's Rule

4. Determinants of Maps

5. The Characteristic Polynomial

6. The Minimal Polynomial

7. Universal Bilinear Functions

8. Tensor Products

9. Exact Sequences

10. Identities on Tensor Products

11. Change of Rings

12. Algebras

CHAPTER X Bilinear and Quadratic Forms

1. Bilinear Forms

2. Symmetric Matrices

3. Quadratic Forms

4. Real Quadratic Fonns

5. Inner Products

6. Orthonormal Bases

7. Orthogonal Matrices

8. The Principal Axis Theorem

9. Unitary Spaces

10. Normal Matrices

CHAPTER XI Similar Matrices and Finite Abelian Groups

1. Noetherian Modules

2. Cyclic Modules

3. Torsion Modules

4. The Rational Canonical Form for Matrices

5. Primary Modules

6. Free Modules

7. Equivalence of Matrices

8. The Calculation of Invariant Factors

CHAPTER XII Structure of Groups

1. Isomorphism Theorems

2. Group Extensions

3. Characteristic Subgroups

4. Conjugate Classes

5. The Sylow Theorems

6. Nilpotent Groups

7. Solvable Groups

8. The JordanHolder Theorem

9. Simplicity of An

CHAPTER XIII Galois Theory

1. Quadratic and Cubic Equations

2. Algebraic and Transcendental Elements

3. Degrees

4. Ruler and Compass

5. Splitting Fields

6. Galois Groups of Polynomials

7. Separable Polynomials

8. Finite Fields

9. Normal Extensions

10. The Fundamental Theorem

11. The Solution of Equations by Radicals

CHAPTER XIV Lattices

1. Posets: Duality Principle

2. Lattice Identities

3. Sublattices and Products of Lattices

4. Modular Lattices

5. JordanHolderDedekind Theorem

6. Distributive Lattices

7. Rings of Sets

8. Boolean Algebras

9. Free Boolean Algebras

CHAPTER XV Categories and Adjoint Functors

1. Categories

2. Functors

3. Contravariant Functors

4. Natural Transformations

5. Representable Functors and Universal Elements

6. Adjoint Functors

CHAPTER XVI Multilinear Algebra

1. Iterated Tensor Products

2. Spaces of Tensors

3. Graded Modules

4. Graded Algebras

5. The Graded Tensor Algebra

6. The Exterior Algebra of a Module

7. Determinants by Exterior Algebra

8. Subspaces by Exterior Algebra

9. Duality in Exterior Algebra

10. Alternating Forms and SkewSymmetric Tensors

APPENDIX Affine and Projective Spaces

1. The Affine Line

2. Affine Spaces

3. The Affine Group

4. Affine Subspaces

5. Biaffine and Quadratic Functionals

6. Euclidean Spaces

7. Euclidean Quadrics

8. Projective Spaces

9. Projective Quadrics

10. Affine and Projective Spaces

Bibliography

Index To the Appendix

Index

Back Cover


Additional Material

Reviews

Nearly every ten years there seems to arrive a new edition of this now classical book the review of which the reviewer hardly can improve. The main advantage of the authors had been the introduction of thoroughly categorical concepts into algebra.
Zentralblatt MATH 
The book is clearly written, beautifully organized, and has an excellent and wideranging supply of exercises ... contains ample material for a fullyear course on modern algebra at the undergraduate level.
Mathematical Reviews


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 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach—emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s—was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
Undergraduates and graduate students interested in algebra.

Front Cover

Preface to the Third Edition

From the Preface to the First Edition

From the Preface to the Second Edition

Contents

List of Symbols

CHAPTER I Sets, Functions, and Integers

1. Sets

2. Functions

3. Relations and Binary Operations

4. The Natural Numbers

5. Addition and Multiplication

6. Inequalities

7. The Integers

8. The Integers Modulo n

9. Equivalence Relations and Quotient Sets

10. Morphisms

11. Semigroups and Monoids

CHAPTER II Groups

1. Groups and Symmetry

2. Rules of Calculation

3. Cyclic Groups

4. Subgroups

5. Defining Relations

6. Symmetric and Alternating Groups

7. Transformation Groups

8. Cosets

9. Kernel and Image

10. Quotient Groups

CHAPTER III Rings

1. Axioms for Rings

2. Constructions for Rings

3. Quotient Rings

4. Integral Domains and Fields

5. The Field of Quotients

6. Polynomials

7. Polynomials as Functions

8. The Division Algorithm

9. Principal Ideal Domains

10. Unique Factorization

11. Prime Fields

12. The Euclidean Algorithm

13. Commutative Quotient Rings

CHAPTER IV Universal Constructions

1. Examples of Universals

2. Functors

3. Universal Elements

4. Polynomials in Several Variables

5. Categories

6. Posets and Lattices

7. Contravariance and Duality

8. The Category of Sets

9. The Category of Finite Sets

CHAPTER V Modules

1. Sample Modules

2. Linear Transformations

3. Submodules

4. Quotient Modules

5. Free Modules

6. Biproducts

7. Dual Modules

CHAPTER VI Vector Spaces

1. Bases and Coordinates

2. Dimension

3. Constructions for Bases

4. Dually Paired Vector Spaces

5. Elementary Operations

6. Systems of Linear Equations

CHAPTER VII Matrices

1. Matrices and Free Modules

2. Matrices and Biproducts

3. The Matrix of a Map

4. The Matrix of a Composite

5. Ranks of Matrices

6. Invertible Matrices

7. Change of Bases

8. Eigenvectors and Eigenvalues

CHAPTER VIII Special Fields

1. Ordered Domains

2. The Ordered Field Q

3. Polynomial Equations

4. Convergence in Ordered Fields

5. The Real Field R

6. Polynomials over R

7. The Complex Plane

8. The Quaternions

9. Extended Formal Power Series

10. Valuations and padic Numbers

CHAPTER IX Determinants and Tensor Products

1. Multilinear and Alternating Functions

2. Determinants of Matrices

3. Cofactors and Cramer's Rule

4. Determinants of Maps

5. The Characteristic Polynomial

6. The Minimal Polynomial

7. Universal Bilinear Functions

8. Tensor Products

9. Exact Sequences

10. Identities on Tensor Products

11. Change of Rings

12. Algebras

CHAPTER X Bilinear and Quadratic Forms

1. Bilinear Forms

2. Symmetric Matrices

3. Quadratic Forms

4. Real Quadratic Fonns

5. Inner Products

6. Orthonormal Bases

7. Orthogonal Matrices

8. The Principal Axis Theorem

9. Unitary Spaces

10. Normal Matrices

CHAPTER XI Similar Matrices and Finite Abelian Groups

1. Noetherian Modules

2. Cyclic Modules

3. Torsion Modules

4. The Rational Canonical Form for Matrices

5. Primary Modules

6. Free Modules

7. Equivalence of Matrices

8. The Calculation of Invariant Factors

CHAPTER XII Structure of Groups

1. Isomorphism Theorems

2. Group Extensions

3. Characteristic Subgroups

4. Conjugate Classes

5. The Sylow Theorems

6. Nilpotent Groups

7. Solvable Groups

8. The JordanHolder Theorem

9. Simplicity of An

CHAPTER XIII Galois Theory

1. Quadratic and Cubic Equations

2. Algebraic and Transcendental Elements

3. Degrees

4. Ruler and Compass

5. Splitting Fields

6. Galois Groups of Polynomials

7. Separable Polynomials

8. Finite Fields

9. Normal Extensions

10. The Fundamental Theorem

11. The Solution of Equations by Radicals

CHAPTER XIV Lattices

1. Posets: Duality Principle

2. Lattice Identities

3. Sublattices and Products of Lattices

4. Modular Lattices

5. JordanHolderDedekind Theorem

6. Distributive Lattices

7. Rings of Sets

8. Boolean Algebras

9. Free Boolean Algebras

CHAPTER XV Categories and Adjoint Functors

1. Categories

2. Functors

3. Contravariant Functors

4. Natural Transformations

5. Representable Functors and Universal Elements

6. Adjoint Functors

CHAPTER XVI Multilinear Algebra

1. Iterated Tensor Products

2. Spaces of Tensors

3. Graded Modules

4. Graded Algebras

5. The Graded Tensor Algebra

6. The Exterior Algebra of a Module

7. Determinants by Exterior Algebra

8. Subspaces by Exterior Algebra

9. Duality in Exterior Algebra

10. Alternating Forms and SkewSymmetric Tensors

APPENDIX Affine and Projective Spaces

1. The Affine Line

2. Affine Spaces

3. The Affine Group

4. Affine Subspaces

5. Biaffine and Quadratic Functionals

6. Euclidean Spaces

7. Euclidean Quadrics

8. Projective Spaces

9. Projective Quadrics

10. Affine and Projective Spaces

Bibliography

Index To the Appendix

Index

Back Cover

Nearly every ten years there seems to arrive a new edition of this now classical book the review of which the reviewer hardly can improve. The main advantage of the authors had been the introduction of thoroughly categorical concepts into algebra.
Zentralblatt MATH 
The book is clearly written, beautifully organized, and has an excellent and wideranging supply of exercises ... contains ample material for a fullyear course on modern algebra at the undergraduate level.
Mathematical Reviews