CONTENTS
Preface = xi
Preliminaries = 1
0.1 Basics = 1
0.2 Properties of the Integers = 4
0.3 Z／n Z : The Integers Modulo n = 8
Part Ⅰ - GROUP THEORY = 13
Chapter 1 Introduction to Groups = 16
1.1 Basic Axioms and Examples = 16
1.2 Dihedral Groups = 23
1.3 Symmetric Groups = 29
1.4 Matrix Groups = 34
1.5 The Quaternion Group = 36
1.6 Homomorphisms and Isomorphisms = 36
1.7 Group Actions = 41
Chapter 2 Subgroups = 46
2.1 Definition and Examples = 46
2.2 Centralizers and Normalizers, Stabilizers and Kernels = 49
2.3 Cyclic Groups and Cyclic Subgroups = 54
2.4 Subgroups Generated by Subsets of a Group = 61
2.5 The Lattice of Subgroups of a Group = 66
Chapter 3 Quotient Groups and Homomorphisms = 73
3.1 Definitions and Examples = 73
3.2 More on Cosets and Lagrange's Theorem = 89
3.3 The Isomorphism Theorems = 97
3.4 Composition Series and the H$$\ddot o$$lder Program = 101
3.5 Transpositions and the Alternating Group = 106
Chapter 4 Group Actions = 112
4.1 Group Actions and Permutation Representations = 112
4.2 Groups Acting on Themselves by Left Multiplication - Cayley's Theorem = 118
4.3 Groups Acting on Themselves by Conjugation - The Class Equation = 122
4.4 Automorphisms = 133
4.5 The Sylow Theorems = 139
4.6 The Simplicity of $$A_n$$ = 149
Chapter 5 Direct and Semidirect Products and Abelian Groups = 152
5.1 Direct Products = 152
5.2 The Fundamental Theorem of Finitely Generated Abelian Groups = 158
5.3 Table of Groups of Small Order = 167
5.4 Recognizing Direct Products = 169
5.5 Semidirect Products = 175
Chapter 6 Further Topics in Group Theory = 188
6.1 p-groups, Nilpotent Groups, and Solvable Groups = 188
6.2 Applications in Groups of Medium Order = 201
6.3 A Word on Free Groups = 215
Part Ⅱ - RING THEORY = 222
Chapter 7 Introduction to Rings = 223
7.1 Basic Definitions and Examples = 223
7.2 Examples : Polynomial Rings, Matrix Rings, and Group Rings = 233
7.3 Ring Homomorphisms an Quotient Rings = 239
7.4 Properties of Ideals = 251
7.5 Rings of Fractions = 260
7.6 The Chinese Remainder Theorem = 265
Chapter 8 Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains = 270
8.1 Euclidean Domains = 270
8.2 Principal Ideal Domains(P.I.D.s) = 279
8.3 Unique Factorization Domains(U.F.D.s) = 283
Chapter 9 Polynomial Rings = 295
9.1 Definitions and Basic Properties = 295
9.2 Polynomial Rings over Fields Ⅰ = 299
9.3 Polynomial Rings that are Unique Factorization Domains = 303
9.4 Irreducibility Criteria = 307
9.5 Polynomial Rings over Fields Ⅱ = 313
9.6 Polynomials in Several Variables over a Field and Gr$$\ddot o$$bner Bases = 315
Part Ⅲ - MODULES AND VECTOR SPACES = 336
Chapter 10 Introduction to Module Theory = 337
10.1 Basic Definitions and Examples = 337
10.2 Quotient Modules and Module Homomorphisms = 345
10.3 Generation of Modules, Direct Sums, and Free Modules = 351
10.4 Tensor Products of Modules = 359
10.5 Exact Sequences - Projective, Injective, and Flat Modules = 378
Chapter 11 Vector Spaces = 408
11.1 Definitions and Basic Theory = 408
11.2 The Matrix of a Linear Transformation = 415
11.3 Dual Vector Spaces = 431
11.4 Determinants = 435
11.5 Tensor Algebras, Symmetric and Exterior Algebras = 441
Chapter 12 Modules over Principal Ideal Domains = 456
12.1 The Basic Theory = 458
12.2 The Rational Canonical Form = 472
12.3 The Jordan Canonical Form = 491
Part Ⅳ - FIELD THEORY AND GALOIS THEORY = 509
Chapter 13 Field Theory = 510
13.1 Basic Theory of Field Extensions = 510
13.2 Algebraic Extensions = 520
13.3 Classical Straightedge and Compass Constructions = 531
13.4 Splitting Fields and Algebraic Closures = 536
13.5 Separable and Inseparable 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
14.7 Solvable and Radical Extensions : Insolvability of the Quintic = 625
14.8 Computation of Galois Groups over Q = 640
14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups = 645
Part Ⅴ - AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA = 655
Chapter 15 Commutative Rings and Algebraic Geometry = 656
15.1 Noetherian Rings and Affine Algebraic Sets = 656
15.2 Radicals and Affine Varieties = 673
15.3 Integral Extensions and Hilbert's Nullstellensatz = 691
15.4 Localization = 706
15.5 The Prime Spectrum of a Ring = 731
Chapter 16 Artinian Rings, Discrete Valuation Rings, and Dedekind Domains = 750
16.1 Artinian Rings = 750
16.2 Discrete Valuation Rings = 755
16.3 Dedekind Domains = 764
Chapter 17 Introduction to Homological Algebra and Group Cohomology = 776
17.1 Introduction to Homological Algebra-Ext and Tor = 777
17.2 The Cohomology of Groups = 798
17.3 Crossed Homomorphisms and H¹(G, A) = 814
17.4 Group Extensions, Factor Sets and H²(G, A) = 824
Part Ⅵ - INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS = 839
Chapter 18 Representation Theory and Character Theory = 840
18.1 Linear Actions and Modules over Group Rings = 840
18.2 Wedderburn's Theorem and Some Consequences = 854
18.3 Character Theory and the Orthogonality Relations = 864
Chapter 19 Examples and Applications of Character Theory = 880
19.1 Characters of Groups of Small Order = 880
19.2 Theorems of Burnside and Hall = 886
19.3 Introduction to the Theory of Induced Characters = 892
Appendix Ⅰ : Cartesian Products and Zorn's Lemma = 905
Appendix Ⅱ : Category Theory = 911
Index = 919
