Abstract algebra / David S. Dummit, Richard M. Foote.

Format:

Book

Author/Creator:

Dummit, David Steven.

Contributor:

Foote, Richard M., 1950-

Rosengarten Family Fund.

Language:

English

Subjects (All):

Algebra, Abstract.

Physical Description:

xii, 932 pages : illustrations ; 25 cm

Edition:

Third edition.

Place of Publication:

Hoboken, NJ : Wiley, [2004]

Summary:

Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. * The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible.

Contents:

0.2 Properties of the Integers 4

0.3 Z / n Z: The Integers Modulo n 8

Part I Group Theory 13

1.2 Dihedral Groups 23

1.3 Symmetric Groups 29

1.4 Matrix Groups 34

1.5 The Quaternion Group 36

1.6 Homomorphisms and Isomorphisms 36

1.7 Group Actions 41

Chapter 2 Subgroups 46

2.2 Centralizers and Normalizers, Stabilizers and Kernels 49

2.3 Cyclic Groups and Cyclic Subgroups 54

2.4 Subgroups Generated by Subsets of a Group 61

2.5 The Lattice of Subgroups of a Group 66

Chapter 3 Quotient Groups and Homomorphisms 73

3.2 More on Cosets and Lagrange's Theorem 89

3.3 The Isomorphism Theorems 97

3.4 Composition Series and the Holder Program 101

3.5 Transpositions and the Alternating Group 106

Chapter 4 Group Actions 112

4.1 Group Actions and Permutation Representations 112

4.2 Groups Acting on Themselves by Left Multiplication

Cayley's Theorem 118

4.3 Groups Acting on Themselves by Conjugation

The Class Equation 122

4.4 Automorphisms 133

4.5 The Sylow Theorems 139

4.6 The Simplicity of A[subscript n] 149

Chapter 5 Direct and Semidirect Products and Abelian Groups 152

5.1 Direct Products 152

5.2 The Fundamental Theorem of Finitely Generated Abelian Groups 158

5.3 Table of Groups of Small Order 167

5.4 Recognizing Direct Products 169

5.5 Semidirect Products 175

Chapter 6 Further Topics in Group Theory 188

6.1 p-groups, Nilpotent Groups, and Solvable Groups 188

6.2 Applications in Groups of Medium Order 201

6.3 A Word on Free Groups 215

Part II Ring Theory 222

Chapter 7 Introduction to Rings 223

7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings 233

7.3 Ring Homomorphisms an Quotient Rings 239

7.4 Properties of Ideals 251

7.5 Rings of Fractions 260

7.6 The Chinese Remainder Theorem 265

Chapter 8 Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains 270

8.1 Euclidean Domains 270

8.2 Principal Ideal Domains (P.I.D.s) 279

8.3 Unique Factorization Domains (U.F.D.s) 283

Chapter 9 Polynomial Rings 295

9.1 Definitions and Basic Properties 295

9.2 Polynomial Rings over Fields I 299

9.3 Polynomial Rings that are Unique Factorization Domains 303

9.4 Irreducibility Criteria 307

9.5 Polynomial Rings over Fields II 313

9.6 Polynomials in Several Variables over a Field and Grobner Bases 315

Part III Modules and Vector Spaces 336

Chapter 10 Introduction to Module Theory 337

10.2 Quotient Modules and Module Homomorphisms 345

10.3 Generation of Modules, Direct Sums, and Free Modules 351

10.4 Tensor Products of Modules 359

10.5 Exact Sequences

Projective, Injective, and Flat Modules 378

Chapter 11 Vector Spaces 408

11.1 Definitions and Basic Theory 408

11.2 The Matrix of a Linear Transformation 415

11.3 Dual Vector Spaces 431

11.4 Determinants 435

11.5 Tensor Algebras, Symmetric and Exterior Algebras 441

Chapter 12 Modules over Principal Ideal Domains 456

12.1 The Basic Theory 458

12.2 The Rational Canonical Form 472

12.3 The Jordan Canonical Form 491

Part IV Field Theory and Galois Theory 509

Chapter 13 Field Theory 510

13.1 Basic Theory of Field Extensions 510

13.2 Algebraic Extensions 520

13.3 Classical Straightedge and Compass Constructions 531

13.4 Splitting Fields and Algebraic Closures 536

13.5 Separable and Inseparable Extensions 545

13.6 Cyclotomic Polynomials and Extensions 552

Chapter 14 Galois Theory 558

14.2 The Fundamental Theorem of Galois Theory 567

14.3 Finite Fields 585

14.4 Composite Extensions and Simple Extensions 591

14.5 Cyclotomic Extensions and Abelian Extensions over Q 596

14.6 Galois Groups of Polynomials 606

14.7 Solvable and Radical Extensions: Insolvability of the Quintic 625

14.8 Computation of Galois Groups over Q 640

14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups 645

Part V An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra 655

Chapter 15 Commutative Rings and Algebraic Geometry 656

15.1 Noetherian Rings and Affine Algebraic Sets 656

15.2 Radicals and Affine Varieties 673

15.3 Integral Extensions and Hilbert's Nullstellensatz 691

15.4 Localization 706

15.5 The Prime Spectrum of a Ring 731

Chapter 16 Artinian Rings, Discrete Valuation Rings, and Dedekind Domains 750

16.1 Artinian Rings 750

16.2 Discrete Valuation Rings 755

16.3 Dedekind Domains 764

Chapter 17 Introduction to Homological Algebra and Group Cohomology 776

17.1 Introduction to Homological Algebra

Ext and Tor 777

17.2 The Cohomology of Groups 798

17.3 Crossed Homomorphisms and H[superscript 1](G, A) 814

17.4 Group Extensions, Factor Sets and H[superscript 2](G, A) 824

Part VI Introduction to the Representation Theory of Finite Groups 839

Chapter 18 Representation Theory and Character Theory 840

18.1 Linear Actions and Modules over Group Rings 840

18.2 Wedderburn's Theorem and Some Consequences 854

18.3 Character Theory and the Orthogonality Relations 864

Chapter 19 Examples and Applications of Character Theory 880

19.1 Characters of Groups of Small Order 880

19.2 Theorems of Burnside and Hall 886

19.3 Introduction to the Theory of Induced Characters 892

Appendix I Cartesian Products and Zorn's Lemma 905

Appendix II Category Theory 911.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

0471433349

OCLC:

52559229

Online:

Publisher description