An Introduction to Abstract Algebra

Description

Title Preface Contents Table of contents : 1. Introduction . . . 1 1-1 Sets. 1 1-2 Mappings. 5 1-3 Binary Operations. 10 1-4 Cartesian Product. 14 1-5 Equivalence Relations. 15 1-6 The Integers. 17 1-7 Finite Induction. 20 1-8 Properties of the Integers. 23 1- 9 Matrices. 26 2. Groups . . . 29 2- 1 Definitions and Elementary Properties. 29 2-2 Subgroups. 32 2-3 Cosets. 35 2-4 Normal Subgroups. 38 2-5 Factor Groups. 40 2-6 Examples of Finite Groups. 42 2-7 Homomorphisms of Groups. 47 2-8 Epimorphisms of Groups. 50 2-9 Isomorphisms of Groups. 52 2-10 Cyclic Groups. 54 2- 11 Permutation Groups. 56 3. Vector Spaces and Linear Transformations . . . 64 3- 1 Definition and Examples. 65 3-2 Subspaces. 69 3-3 Basis and Dimension of a Vector Space. 71 3-4 Linear Transformations. 76 3-5 Sets of Linear Transformations. 80 3-6 The Matrix of a Linear Transformation. 82 3-7 Change of Basis. 86 3-8 Quotient Spaces. 88 3-9 The Inner Product. 92 3-10 Orthonormal Basis. 95 3-11 Isometries. 99 3-12 Dual Vector Spaces.101 3- 13 Eigenvectors and the Spectral Theorem.103 4. Structure of Groups ... 114 4- 1 Conjugate Classes and Subgroups.114 4-2 The Sylow Theorems.119 4-3 Automorphisms of Groups.124 4-4 Direct Product of Groups.126 4-5 The External Direct Product.129 4-6 Free Abelian Groups.131 4-7 The Basis of a Free Abelian Group.135 4-8 Finitely Generated Abelian Groups.138 4-9 Subnormal and Composition Series.144 4-10 Nilpotent Groups.149 4- 11 Groups with Operators.151 5. Rings ... 156 5- 1 Definitions.156 5-2 Subrings and Ideals.162 5-3 The Characteristic of a Ring.166 5-4 Ring Homomorphisms.168 5-5 The Ring of Endomorphisms of an Abelian Group.172 5-6 Extensions of Rings.174 5-7 Polynomial Rings.178 5-8 Properties of Polynomial Rings.184 5- 9 Polynomial Rings in Two Variables.187 6. Factorization and Ideals ... 790 6- 1 Primes.190 6-2 Unique Factorization Domains.192 6-3 Principal Ideal Domains.193 6-4 The Unique Factorization Domain D[x].197 6-5 Euclidean Domains.202 6-6 Prime Ideals and Maximal Ideals.204 6-7 Simple Operations with Ideals.207 6-8 Quadratic Domains.209 6- 9 Basis Theorem.211 7. Modules ... 216 7- 1 Definitions.216 7-2 Module Homomorphisms.220 7-3 Finitely Generated Modules.222 7-4 Direct Sums of Modules.223 7-5 Representations of Rings.225 7-6 The Chain Conditions.226 7-7 Primary Decompositions in Noetherian Rings.230 7-8 Free Modules.234 7-9 Modules over a Principal Ideal Domain.239 7-10 Projective Modules.242 7-11 The Tensor Product.243 7-12 Module Homomorphisms on Tensor Products.247 7-13 Tensor Product of a Finite Number of Modules.249 7- 14 Tensor Product of a Direct Sum.251 8. Algebras . . . 256 8- 1 Algebras and Ideals.256 8-2 Graded Modules.261 8-3 Graded Algebras.264 8-4 Tensor Algebras.266 8-5 Exterior Algebras.268 8-6 Tensor Products of Algebras.271 8-7 Applications.272 8-8 Graded Tensor Algebras.275 8-9 Lie Algebras.277 8-10 Tensors.279 8- 11 Tensor Spaces.280 9. Field Theory and Galois Theory . . . 284 9- 1 Structure of Fields.284 9- 2 Field Extensions.285 9-3 Algebraic Extensions.288 9-4 Properties of An Algebraic Extension.292 9-5 Splitting Fields and Normal Extensions.294 9-6 Roots of Unity.299 9- 7 Galois Theory.200 9-8 Extensions by Radicals.205 9-9 The General Equation of Degree n.307 9-10 Cohomology of a Galois Group.311 10. Homological Algebra ... 313 10- 1 The Splitting of Exact Sequences.314 10-2 Projective Modules.317 10-3 Injective Modules.321 10-4 A Tensor Product.323 10-5 Categories and Functors.324 10- 6 Bifunctors.^2% 10-7 Universal Elements.330 10- 8 Homology Groups.331 11. Elementary Structure Theory of Rings ... 341 11- 1 The Radical.341 11-2 Primitive Rings.347 11-3 The d.c.c. and Simple Rings.351 11-4 The Density Theorem.353 11-5 The Principal Structure Theorem for Semisimple Rings with the d.c.c.358 Bibliography . . . 363 Index . . . 365