An Introduction to Abstract Algebra

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