Algebra: Third Edition

Item Successfully Added to Cart

An error was encountered while trying to add the item to the cart. Please try again.

Continue Shopping

Go to Checkout

Please make all selections above before adding to cart

Share this page via the icons above, or by copying the link below:

Copy To Clipboard

Successfully Copied!

Algebra: Third Edition

Saunders Mac Lane

Garrett Birkhoff

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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

Add to cart

Click above image for expanded view

Algebra: Third Edition

Saunders Mac Lane

Garrett Birkhoff

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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

Add to cart

Book Details
Table of Contents
Additional Material
Reviews
Requests

AMS Chelsea Publishing

Volume: 330; 1988; 626 pp

MSC: 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.

Readership

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

Review Copy – for publishers of book reviews

Desk Copy – for instructors who have adopted an AMS textbook for a course

Examination Copy – for faculty considering an AMS textbook for a course

Permission – for use of book, eBook, or Journal content

Accessibility – to request an alternate format of an AMS title

Please select which format for which you are requesting permissions.