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. Frequently Used Notation 3 Contents 7 Preface 12 Preliminaries 14 0.1 Basics 14 0.2 Properties of the Integers 17 0.3 Z/nZ: The Integers Modulo n 21 Part I - Group Theory 26 Chapter 1 - Introduction to Groups 29 1.1 - Basic Axioms and Examples 29 1.2 - Dihedral Groups 36 1.3 - Symmetric Groups 42 1.4 - Matrix Groups 47 1.5 - The Quaternion Group 49 1.6 - Homomorphisms and Isomorphisms 49 1.7 - Group Actions 54 Chapter 2 - Subgroups 59 2.1 - Definition and Examples 59 2.2 - Centralizers and Normalizers, Stabilizers and Kernels 62 2.3 - Cyclic Groups and Cyclic Subgroups 67 2.4 - Subgroups Generated by Subsets of a Group 74 2.5 - The Lattice of Subgroups of a Group 79 Chapter 3 - Quotient Groups and Homomorphisms 86 3.1 - Definitions and Examples 86 3.2 - More on Cosets and Lagrange's Theorem 102 3.3 - The Isomorphism Theorems 110 3.4 - Composition Series and the Hölder Program 114 3.5 - Transpositions and the Alternating Group 119 Chapter 4 - Group Actions 125 4.1 - Group Actions and Permutation Representations 125 4.2 - Groups Acting on Themselves by Left Multiplication—Cayley's Theorem 131 4.3 - Groups Acting on Themselves by Conjugation—The Class Equation 135 4.4 - Automorphisms 146 4.5 - The Sylow Theorems 152 4.6 - The Simplicity of A_n 162 Chapter 5 - Direct and Semidirect Products and Abelian Groups 165 5.1 - Direct Products 165 5.2 - The Fundamental Theorem of Finitely Generated Abelian Groups 171 5.3 - Table of Groups of Small Order 180 5.4 - Recognizing Direct Products 182 5.5 - Semidirect Products 188 Chapter 6 - Further Topics in Group Theory 201 6.1 - p-groups, Nilpotent Groups, and Solvable Groups 201 6.2 - Applications in Groups of Medium Order 214 6.3 - A Word on Free Groups 228 Part II - Ring Theory 235 Chapter 7 - Introduction to Rings 236 7.1 - Basic Definitions and Examples 236 7.2 - Examples: Polynomial Rings, Matrix Rings, and Group Rings 246 7.3 - Ring Homomorphisms and Quotient Rings 252 7.4 - Properties of Ideals 264 7.5 - Rings of Fractions 273 7.6 - The Chinese Remainder Theorem 278 Chapter 8 - Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains 283 8.1 - Euclicdean Domains 283 8.2 - Principal Ideal Domains (P.I.D.s) 292 8.3 - Unique Factorization Domains (U.F.D.s) 296 Chapter 9 - Polynomial Rings 308 9.1 - Definitions and Basic Properties 308 9.2 - Polynomial Rings over Fields I 312 9.3 - Polynomial Rings that are Unique Factorization Domains 316 9.4 - Irriducibility Criteria 320 9.5 - Polynomial Rings over Fields II 326 9.6 - Polynomials in Several Variables over a Field and Gröbner Bases 328 Part III - Modules and Vector Spaces 349 Chapter 10 - Introduction to Module Theory 350 10.1 - Basic Definitions and Examples 350 10.2 - Quotient Modules and Module Homomorphisms 358 10.3 - Generation of Modules, Direct Sums, and Free Modules 364 10.4 - Tensor Products of Modules 372 10.5 - Exact Sequences—Projective, Injective, and Flat Modules 391 Chapter 11 - Vector Spaces 421 11.1 - Definitions and Basic Theory 421 11.2 - The Matrix of a Linear Transformation 428 11.3 - Dual Vector Spaces 444 11.4 - Determinants 448 11.5 - Tensor Algebras, Symmetric and Exterior Algebras 454 Chapter 12 - Modules over Principal Ideal Domains 469 12.1 - The Basic Theory 471 12.2 - The Rational Canonical Form 485 12.3 - The Jordan Canonical Form 504 Part IV - Field Theory and Galois Theory 522 Chapter 13 - Field Theory 523 13.1 - Basic Theory of Field Extensions 523 13.2 - Algebraic Extensions 533 13.3 - Classical Straightedge and Compass Constructions 544 13.4 - Splitting Fields and Algebraic Closures 549 13.5 - Separable and Inseparable Extensions 558 13.6 - Cyclotomic Polynomials and Extensions 565 Chapter 14 - Galois Theory 571 14.1 - Basic Definitions 571 14.2 - The Fundamental Theorem of Galois Theory 580 14.3 - Finite Fields 598 14.4 - Composite Extensions and Simple Extensions 604 14.5 - Cyclotomic Extensions and Abelian Extensions over Q 609 14.6 - Galois Groups of Polynomials 619 14.7 - Solvable and Radical Extensions: Insolvability of the Quintic 638 14.8 - Computation of Galois Groups over Q 653 14.9 - Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups 658 Part V - An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra 668 Chapter 15 - Commutative Rings and Algebraic Geometry 669 15.1 - Noetherian Rings and Affine Algebraic Sets 669 15.2 - Radicals and Affine Varieties 686 15.3 - Integral Extensions and Hilber's Nullstellensatz 704 15.4 - Localization 719 15.5 - The Prime Spectrum of a Ring 744 Chapter 16 - Artinian Rings, Discrete Valuation Rings, and Dedekind Domains 763 16.1 - Artinian Rings 763 16.2 - Discrete Valuation Rings 768 16.3 - Dedekind Domains 777 Chapter 17 - Introduction to Homological Algebra and Group Cohomology 789 17.1 - Introduction to homological Algebra—Ext and Tor 790 17.2 - The Cohomology of Groups 811 17.3 - Crossed Homomorphisms and H^1(G,A) 827 17.4 - Group Extensions, Factor Sets, and H^2(G,A) 837 Part VI - Introduction to the Representation Theory of Finite Groups 852 Chapter 18 - Representation Theory and Character Theory 853 18.1 - Linear Actions and Modules over Group Rings 853 18.2 - Wedderburn's Theorem and Some Consequences 867 18.3 - Character Theory and the Orthogonality Relations 877 Chapter 19 - Examples and Applications of Character Theory 893 19.1 - Characters of Groups of Small Order 893 19.2 - Theorems of Burnside and Hall 899 19.3 - Introduction to the Theory of Induced Characters 905 Appendix I - Cartesian Products and Zorn's Lemma 918 Appendix II - Category Theory 924 Index 932 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.