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. Cover Frequently Used Notation Contents Preface to the Third Edition Preliminaries 0.1 Basics 0.2 Properties of the Integers 0.3 Z/nZ: The Integers Modulo n Part I - Group Theory Chapter 1 - Introduction to Groups 1.1 Basic Axioms and Examples 1.2 Dihedral Groups 1.3 Symmetric Groups 1.4 Matrix Groups 1.5 The Quaternion Group 1.6 Homomorphisms and Isomorphisms 1.7 Group Actions Chapter 2 - Subgroups 2.1 Definition and Examples 2.2 Centralizers and Normalizers, Stabilizers and Kernels 2.3 Cyclic Groups and Cyclic Subgroups 2.4 Subgroups Generated by Subsets of a Group 2.5 The Lattice of Subgroups of a Group Chapter 3 - Quotient Groups and Homomorphisms 3.1 Definition and Examples 3.2 More on Cosets and Lagrange's Theorem 3.3 The Isomorphism Theorems 3.4 Composition Series and the Holder Program 3.5 Transpositions and the Alternating Group Chapter 4 - Group Actions 4.1 Group Actions and Permutation Representations 4.2 Groups Acting on Themselves by Left Multiplication - Cayley's Theorem 4.3 Groups Acting on Themselves by Conjugation - The Class Euation 4.4 Automorphisms 4.5 The Sylow Theorems 4.6 The Simplicity of A_n Chapter 5 - Direct and Semidirect Products and Abelian Groups 5.1 Direct Products 5.2 The Fundamental Theorem of Finitely Generated Abelian Groups 5.3 Table of Groups of Small Order 5.4 Recognizing Direct Products 5.5 Semidirect Products Chapter 6 - Further Topics in Group Theory 5.1 p-groups, Nilpotent Groups, and Solvable Groups 5.2 Applications in Groups of Medium Order 5.3 A Word on Free Groups Part II - Ring Theory Chapter 7 - Introduction to Rings 7.1 Basic Definitions and Examples 7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings 7.3 Ring Homomorphisms and Quotient Rings 7.4 Properties of Ideals 7.5 Rings of Fractions 7.6 The Chinese Remainder Theorem Chapter 8 - Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains 8.1 Euclidean Domains 8.2 Principal Ideal Domains (P.I.D.s) 8.3 Unique Factorization Domains (U.F.D.s) Chapter 9 - Polynomial Rings 9.1 Definitions and Basic Properties 9.2 Polynomial Rings over Fields I 9.3 Polynomial Rings that are Unique Factorization Ideals 9.4 Irreducibility Criteria 9.5 Polynomial Rings over Fields II 9.6 Polynomials in Several Variables over a Field and Grobner Bases Part III - Modules and Vector Spaces Chapter 10 - Introduction to Module Theory 10.1 Basic Definitions and Examples 10.2 Quotient Modules and Module Homomorphisms 10.3 Generation of Modules, Direct Sums, and Free Modules 10.4 Tensor Products of Modules 10.5 Exact Sequences - Projective, Injective, and Flat Modules Chapter 11 - Vector Spaces 11.1 Definitions and Basic Theory 11.2 The Matrix of a Linear Transformation 11.3 Dual Vector Spaces 11.4 Determinants 11.5 Tensor Algebras, Symmetric and Exterial Algebras Chapter 12 - Modules over Principal Ideal Domains 12.1 The Basic Theory 12.2 The Rational Canonical Form 12.3 The Jordan Canonical Form Part IV - Field Theory and Galois Theory Chapter 13 - Field Theory 13.1 Basic Theory of Field Extensions 13.2 Algebraic Extensions 13.3 Classical Straightedge and Compass Constructions 13.4 Splitting Fields and Algebraic Closures 13.5 Separable and Inseparable Extensions 13.6 Cyclotomic Polynomials and Extensions Chapter 14 - Galois Theory 14.1 Basic Definitions 14.2 The Fundamental Theorem of Galois Theory 14.3 Finite Fields 14.4 Composite Extensions and Simple Extensions 14.5 Cyclotomic Extensions and Abelian Extensions over Q 14.6 Galois Groups of Polynomials 14.7 Solvable and Radical Extensions: Insolvability of the Quintic 14.8 Computation of Galois Groups over Q 14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups Part V - An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra Chapter 15 - Commutative Rings and Algebraic Geometry 15.1 Noetherian Rings and Affine Algebraic Sets 15.2 Radicals and Affine Varieties 15.3 Integral Extensions and Hilbert's Nullstellensatz 15.4 Localization 15.5 The Prime Spectrum of a Ring Chapter 16 - Artinian Rings, Discrete Valuation Rings, and Dedekind Domains 16.1 Artinian Rings 16.2 Discrete Valuation Rings 16.3 Dedekind Domains Chapter 17 - Introduction to Homological Algebra and Group Cohomology 17.1 Introduction to Homological Algebra - Ext and Tor 17.2 The Cohomology of Groups 17.3 Crossed Homomorphisms and H^1(G,A) 17.4 Group Extensions, Factor Sets and H^2(G,A) Part VI - Introduction to the Representation Theory of Finite Groups Chapter 18 - Representation Theory and Character Theory 18.1 Linear Actions and Modules over Group Rings 18.2 Wedderburn's Theorem and Some Consequences 18.3 Character Theory and the Orthogonality Relations Chapter 19 - Examples and Applications of Character Theory 19.1 Characters of Groups of Small Order 19.2 Theorems of Burnside and Hall 19.3 Introduction to the Theory of Induced Characters Appendix I: Cartesian Products and Zorn's Lemma Appendix II: Category Theory Index 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.