Preface | p. xi |
Preliminaries | p. 1 |
Basics | p. 1 |
Properties of the Integers | p. 4 |
Z / n Z: The Integers Modulo n | p. 8 |
Group Theory | p. 13 |
Introduction to Groups | p. 16 |
Basic Axioms and Examples | p. 16 |
Dihedral Groups | p. 23 |
Symmetric Groups | p. 29 |
Matrix Groups | p. 34 |
The Quaternion Group | p. 36 |
Homomorphisms and Isomorphisms | p. 36 |
Group Actions | p. 41 |
Subgroups | p. 46 |
Definition and Examples | p. 46 |
Centralizers and Normalizers, Stabilizers and Kernels | p. 49 |
Cyclic Groups and Cyclic Subgroups | p. 54 |
Subgroups Generated by Subsets of a Group | ...p. 61 |
The Lattice of Subgroups of a Group | p. 66 |
Quotient Groups and Homomorphisms | p. 73 |
Definitions and Examples | p. 73 |
More on Cosets and Lagrange's Theorem | p. 89 |
The Isomorphism Theorems | p. 97 |
Composition Series and the Holder Program | p. 101 |
Transpositions and the Alternating Group | p. 106 |
Group Actions | p. 112 |
Group Actions and Permutation Representations | p. 112 |
Groups Acting on Themselves by Left Multiplication--Cayley's Theorem | p. 118 |
Groups Acting on Themselves by Conjugation--The Class Equation | p. 122 |
Automorphisms | p. 133 |
The Sylow Theorems | p. 139 |
The Simplicity of A[subscript n] | p. 149 |
Direct and Semidirect Products and Abelian Groups | p. 152 |
Direct Products | p. 152 |
The Fundamental Theorem of Finitely Generated Abelian Groups | p. 158 |
Table of Groups of Small Order | p. 167 |
Recognizing Direct Products | p. 169 |
Semidirect Products | p. 175 |
Further Topics in Group Theory | p. 188 |
p-groups, Nilpotent Groups, and Solvable Groups | p. 188 |
Applications in Groups of Medium Order | p. 201 |
A Word on Free Groups | p. 215 |
Ring Theory | p. 222 |
Introduction to Rings | p. 223 |
Basic Definitions and Examples | p. 223 |
Examples: Polynomial Rings, Matrix Rings, and Group Rings | p. 233 |
Ring Homomorphisms an Quotient Rings | p. 239 |
Properties of Ideals | p. 251 |
Rings of Fractions | p. 260 |
The Chinese Remainder Theorem | p. 265 |
Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains | p. 270 |
Euclidean Domains | p. 270 |
Principal Ideal Domains (P.I.D.s) | p. 279 |
Unique Factorization Domains (U.F.D.s) | p. 283 |
Polynomial Rings | p. 295 |
Definitions and Basic Properties | p. 295 |
Polynomial Rings over Fields I | p. 299 |
Polynomial Rings that are Unique Factorization Domains | p. 303 |
Irreducibility Criteria | p. 307 |
Polynomial Rings over Fields II | p. 313 |
Polynomials in Several Variables over a Field and Grobner Bases | p. 315 |
Modules and Vector Spaces | p. 336 |
Introduction to Module Theory | p. 337 |
Basic Definitions and Examples | p. 337 |
Quotient Modules and Module Homomorphisms | p. 345 |
Generation of Modules, Direct Sums, and Free Modules | p. 351 |
Tensor Products of Modules | p. 359 |
Exact Sequences--Projective, Injective, and Flat Modules | p. 378 |
Vector Spaces | p. 408 |
Definitions and Basic Theory | p. 408 |
The Matrix of a Linear Transformation | p. 415 |
Dual Vector Spaces | p. 431 |
Determinants | p. 435 |
Tensor Algebras, Symmetric and Exterior Algebras | p. 441 |
Modules over Principal Ideal Domains | p. 456 |
The Basic Theory | p. 458 |
The Rational Canonical Form | p. 472 |
The Jordan Canonical Form | p. 491 |
Field Theory and Galois Theory | p. 509 |
Field Theory | p. 510 |
Basic Theory of Field Extensions | p. 510 |
Algebraic Extensions | p. 520 |
Classical Straightedge and Compass Constructions | p. 531 |
Splitting Fields and Algebraic Closures | p. 536 |
Separable and Inseparable Extensions | p. 545 |
Cyclotomic Polynomials and Extensions | p. 552 |
Galois Theory | p. 558 |
Basic Definitions | p. 558 |
The Fundamental Theorem of Galois Theory | p. 567 |
Finite Fields | p. 585 |
Composite Extensions and Simple Extensions | p. 591 |
Cyclotomic Extensions and Abelian Extensions over Q | p. 596 |
Galois Groups of Polynomials | p. 606 |
Solvable and Radical Extensions: Insolvability of the Quintic | p. 625 |
Computation of Galois Groups over Q | p. 640 |
Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups | p. 645 |
An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra | p. 655 |
Commutative Rings and Algebraic Geometry | p. 656 |
Noetherian Rings and Affine Algebraic Sets | p. 656 |
Radicals and Affine Varieties | p. 673 |
Integral Extensions and Hilbert's Nullstellensatz | p. 691 |
Localization | p. 706 |
The Prime Spectrum of a Ring | p. 731 |
Artinian Rings, Discrete Valuation Rings, and Dedekind Domains | p. 750 |
Artinian Rings | p. 750 |
Discrete Valuation Rings | p. 755 |
Dedekind Domains | p. 764 |
Introduction to Homological Algebra and Group Cohomology | p. 776 |
Introduction to Homological Algebra--Ext and Tor | p. 777 |
The Cohomology of Groups | p. 798 |
Crossed Homomorphisms and H[superscript 1](G, A) | p. 814 |
Group Extensions, Factor Sets and H[superscript 2](G, A) | p. 824 |
Introduction to the Representation Theory of Finite Groups | p. 839 |
Representation Theory and Character Theory | p. 840 |
Linear Actions and Modules over Group Rings | p. 840 |
Wedderburn's Theorem and Some Consequences | p. 854 |
Character Theory and the Orthogonality Relations | p. 864 |
Examples and Applications of Character Theory | p. 880 |
Characters of Groups of Small Order | p. 880 |
Theorems of Burnside and Hall | p. 886 |
Introduction to the Theory of Induced Characters | p. 892 |
Cartesian Products and Zorn's Lemma | p. 905 |
Category Theory | p. 911 |
Index | p. 919 |
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