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Softcover ISBN: | 978-1-4704-7476-8 |
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Softcover ISBN: | 978-1-4704-7476-8 |
Product Code: | CHEL/330.S |
List Price: | $69.00 |
MAA Member Price: | $62.10 |
AMS Member Price: | $62.10 |
eBook ISBN: | 978-1-4704-7603-8 |
Product Code: | CHEL/330.E |
List Price: | $65.00 |
MAA Member Price: | $58.50 |
AMS Member Price: | $58.50 |
Softcover ISBN: | 978-1-4704-7476-8 |
eBook ISBN: | 978-1-4704-7603-8 |
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 p-adic 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 Jordan-Holder 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. Jordan-Holder-Dedekind 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 Skew-Symmetric 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 wide-ranging supply of exercises ... contains ample material for a full-year course on modern algebra at the undergraduate level.
Mathematical Reviews
-
-
RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- 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 p-adic 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 Jordan-Holder 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. Jordan-Holder-Dedekind 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 Skew-Symmetric 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 wide-ranging supply of exercises ... contains ample material for a full-year course on modern algebra at the undergraduate level.
Mathematical Reviews