Abstract Algebra: Third Edition

Main subject categories: • Abstract Algebra • Integers • Group Theory • Ring Theory • Modules • Vector Spaces • Field Theory • Galois Theory • Algebraic Geometry • Homological Algebra • Representation Theory of Finite GroupsThis revision of Dummit and Foote's widely acclaimed introduction to abstract algebra helps students experience the power and beauty that develops from the 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 student's understanding. With this approach, students gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings.The text is designed for a full-year introduction to abstract algebra at the advanced undergraduate or graduate level, but contains substantially more material than would normally be covered in one year. Portions of the book may also be used for various one-semester topics courses in advanced algebra, each of which would provide a solid background for a follow-up course delving more deeply into one of many possible areas: algebraic number theory, algebraic topology, algebraic geometry, representation theory, Lie groups, etc. Preface Contents Preliminaries 1 0.1 0.2 0.3 Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Chapter 2 2.1 2.2 2.3 2.4 2.5 Contents Basics 1 Properties of the Integers 4 Z /n Z : The Integers Modulo n 8 Part I — GROUP THEORY 13 Introduction to Groups 1 6 Basic Axioms and Examples 16 Dihedral Groups 23 Symmetric Groups 29 Matrix Groups 34 The Quaternion Group 36 Homomorphisms and lsomorphisms 36 Group Actions 41 Subgroups 46 Definition and Examples 46 Centralizers and Normalizers, Stabilizers and Kemels 49 Cyclic Groups and Cyclic Subgroups 54 Subgroups Generated by Subsets of a Group 61 The Lattice of Subgroups of a Group 66 Chapter 3 3-1 3.2 3.3 3.4 3.5 Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 Chapter 5 5.1 5.2 5.3 5.4 5.5 Chapter 6 6. 1 6.2 6.3 Chapter 7 7. 1 7.2 7.3 7.4 7.5 7.6 vi Quotient Groups and Homomorphisms 73 Definitions and Examples 73 More on Cosets and Lagrange’s Theorem 89 The lsomorphism Theorems 97 Composition Series and the Holder Program 101 Transpositions and the Alternating Group 106 Group Actions 1 1 2 Group Actions and Permutation Representations 112 Groups Acting on Themselves by Left Multiplication—Cayley’s Theorem 1 18 Groups Acting on Themselves by Conjugation—The Class Equation 122 Automorphisms 133 The Sylow Theorems 139 The Simplicity of An 149 Direct and Semidirect Products and Abelian Groups 1 52 Direct Products 152 The Fundamental Theorem of Finitely Generated Abelian Groups 1 58 Table of Groups of Small Order 167 Recognizing Direct Products 169 Semidirect Products 175 Further Topics in Group Theory 1 88 p-groups, Nilpotent Groups, and Solvable Groups 188 Applications in Groups of Medium Order 201 A Word on Free Groups 21 5 Part ll — RING THEORY 222 Introduction to Rings 223 Basic Definitions and Examples 223 Examples: Polynomial Rings, Matrix Rings, and Group Rings 233 Ring Homomorphisms an Quotient Rings 239 Properties of Ideals 251 Rings of Fractions 260 The Chinese Remainder Theorem 265 Contents Chapter 8 8.1 8.2 8.3 Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains 270 Euclidean Domains 270 Principal Ideal Domains (P.l.D.s) 279 Unique Factorization Domains (U.F.D.s) 283 Polynomial Rings 295 Definitions and Basic Properties 295 Polynomial Rings over Fields I 299 Polynomial Rings that are Unique Factorization Domains 303 lrreducibility Criteria 307 Polynomial Rings over Fields ll 313 Polynomials in Several Variables overa Field and Grobner Bases 31 5 Part Ill — MODULES AND VE(.TOR SPACES 336 Chapter 1 0 1 0.1 1 0.2 1 0.3 1 0.4 1 0.5 Chapter 1 1 11.1 11.2 11.3 11.4 11.5 Chapter 1 2 12.1 12.2 12.3 Contents Introduction to Module Theory 337 Basic Definitions and Examples 337 Quotient Modules and Module Homomorphisms 345 Generation of Modules, Direct Sums, and Free Modules 351 Tensor Products of Modules 359 Exact Sequences—Projective, lnjective, and Flat Modules 378 Vector Spaces 408 Definitions and Basic Theory 408 The Matrix of a Linear Transformation 41 5 Dual Vector Spaces 431 Determinants 435 Tensor Algebras, Symmetric and Exterior Algebras 441 Modules over Principal Ideal Domains 456 The Basic Theory 458 The Rational Canonical Form 472 The Jordan Canonical Form 491 vii Part IV — FIELD THEORY AND GALOIS THEORY 509 Chapter 13 Field Theory 510 13.1 Basic Theory of Field Extensions 51 0 13.2 Algebraic Extensions 520 1 3.3 Classical Straightedge and Compass Constructions 531 13.4 Splitting Fields and Algebraic Closures 536 13.5 Separable and lnseparable Extensions 545 13.6 Cyclotomic Polynomials and Extensions 552 Chapter 14 Galois Theory 558 14.1 Basic Definitions 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 1 4.7 Solvable and Radical Extensions: lnsolvability of the Quintic 625 14.8 Computation of Galois Groups over Q 640 14.9 Transcendental Extensions, lnseparable Extensions, Infinite Galois Groups 645 Part V — AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA 655 Chapter 1 5 Commutative Rings and Algebraic Geometry 656 1 5.1 Noetherian Rings and Affine Algebraic Sets 656 1 5.2 Radicals and Affine Varieties 673 15.3 Integral Extensions and Hilbert’s Nullstellensatz 691 1 5.4 Localization 706 15.5 The Prime Spectrum of a Ring 731 Chapter 1 6 Artinian Rings, Discrete Valuation Rings, and Dedekind Domains 750 16.1 Artinian Rings 750 1 6.2 Discrete Valuation Rings 755 16.3 Dedekind Domains 764 viii Content Chapter 1 7 1 7.1 1 7.2 1 7.3 1 7.4 Introduction to Homological Algebra and Group Cohomology 776 Introduction to Homological Algebra—Ext and Tor 777 The Cohomology of Groups 798 Crossed Homomorphisms and H7(G, A) 814 Group Extensions, Factor Sets and H2(G, A) 824 Part VI — INTRODUCTION TO THE REPRESENTATION Chapter 1 8 1 8.1 1 8.2 1 8.3 Chapter 19 1 9.1 1 9.2 1 9.3 THEORY OF FINITE GROUPS 839 Representation Theory and Character Theory 840 Linear Actions and Modules over Group Rings 840 Wedderburn’s Theorem and Some Consequences 854 Character Theory and the Orthogonality Relations 864 Examples and Applications of Character Theory 880 Characters of Groups of Small Order 880 Theorems of Burnside and Hall 886 Introduction to the Theory of Induced Characters 892 Appendix' I.' Cartesian Products and Zorn’s Lemma 905 Appendix II: Index 919 Preface Preliminaries Basics Properties of the Integers Z/nZ: The Integers Modulo n Group Theory Introduction to Groups Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Subgroups Definitions and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of Subgroups of a Group Quotient Groups and Homomorphisms Definitions and Examples More on Cosets and Lagrange's Theorem The Isomorphism Theorems Composition Series and the Holder Program Transpositions and the Alternating Group Group Actions Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication-Cayley's Theorem Groups Acting on Themselves by Conjugation-The Class Equation Automorphism The Sylow Theorems The Simplicity of An Direct and Semidirect Products and Abelian Groups Direct Products The Fundamental Theorem of Finitely Generated Ableian Groups Table of Groups of Small Order Recognizing Direct Products Semidirect Products Further Topics in Group Theory p-groups, Nilpotent Groups, and Solvable Groups Applications in Groups of Medium Order A Word on Free Groups Ring Theory Introduction to Rings Basic Definitions and Exmaples Examples: Polynomial Rings, Matri Rings, and Group Rings Ring Homomorphisms and Quotinet Rings Properties of Ideals Rings of fractions The Chinese Remainder Theorem Euclidean Domains, Principla Ideal Domains and Unique Factorization Domains Euclidean Domains Principal Ideal Domains (P.I.D.s) Unique Factorization Domains (U.F.D.s) Polynomial Rings Definitions and Basic Properties Polynomial Rings over Fields I Polynomial Rings that are Unique Factorization Domains Irreducibility Criteria Polynomial rings over Fields II Polynomials in Several Variables over a Field and Grobner Bases Modules and Vector Spaces Introduction to Module Theory Basic Definitions and Examples Quotient Modules and Module Homomorphisms Generation of Modules, Direct Sums, and Free Modules Tensor Product of Modules Exact Sequences - Projective, Injective, and Flat Modules vector Spaces Definitions and Basic Theory The Matrix of a Linear Transformation Dual Vector Spaces Determinants Tensor Algebras, Symmetric and Exterior Algebras Modules over Principal Ideal Domains The Basic Theory The Rational Canonical From The Jordan Canonical From Field Theory Basic Theory of field Extensions Algebraic extensions Classical Straightedge and Compass Constructions Splitting Fields and Algebraic Closures Separable and Inseparable Extensions Cyclotomic Polynomials and Extensions Galios Theory Basic Definitions The Fundamental Theorem of Galios Theory Finite Fields Composite Extensions and Simple Extensions Cyclotomic Extensions and Albelian Extensions over Q Galois Groups of Polynomials Solvable and Radical Extensions: Insolvability of the Quintic Computation of the Galois Groups over Q Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups An Introduction to Comutative Rings, Algebraic Geometry, and Homological Algebra Commutative Rings and Algebraic Geometry Noetherian Rings and Affine Algebraic Sets Radicals and Affine Varieties Integral Extensions and Hilbert's Nullstellensatz Localization The Prime Spectrum of a Ring Artinian Rings, Discrete Valuation Rings, and Dedekind Domains Artinian Rings Discrete Valuation Rings Dedekind Domains Introduction to Homological Algebra and Group Cohomology Introduction to Homological Algebra - Ext and Tor The Cohomology of Groups Crossed Homomorphisms and H1(G, A) Group Extensions, Factor Sets and H2(G, A) Introduction to the Representation Theory of Finite Groups Representation Theory and Character Theory Linear Actions and Modules over Group Rings Wedderburn's Theorem and Some Consequences Character Theory and the Orthogonality Relations Examples and Applications of Character Theory Characters of Groups of Small Order Theorems of Burnside and Hall Introduction to the Theory of Induced Characters Cartesian Products and Zorn's Lemma Category Theory Index 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. Part. 1. Group Theory: Introduction To Groups -- Subgroups -- Quotient Groups And Homomorphisms -- Group Actions -- Direct And Semidirect Products And Abelian Groups -- Further Topics In Group Theory -- Part 2. Ring Theory: Introduction To Rings -- Euclidean Domains, Principal Ideal Domains And Unique Factorization Domains -- Polynomial Rings -- Part 3. Modules And Vector Spaces: Introduction To Module Theory -- Vector Spaces -- Modules Over Principal Ideal Domains -- Part. 4. Field Theory And Galois Theory: Field Theory -- Galois Theory -- Part 5. An Introduction To Commutative Rings, Algebraic Geometry, And Homological Algebra: Commutative Rings And Algebraic Geometry -- Artinian Rings, Discrete Valuation Rings, And Dedekind Domains -- Introduction To Homological Algebra And Group Cohomology -- Part 6. Introduction To The Representation Theory Of Finite Groups: Representation Theory And Character Theory -- Examples And Applications Of Character Theory. David S. Dummit, Richard M. Foote. Includes Index. 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.