| Preface | p. v |
| Preliminaries | p. 1 |
| Basic Ideas of Set Theory | p. 2 |
| Functions | p. 7 |
| Equivalence Relations and Partitions | p. 11 |
| A Note on Natural Numbers | p. 14 |
| Review Exercises | p. 16 |
| Algebraic Structure of Numbers | p. 17 |
| The Set of Integers | p. 18 |
| Congruences of Integers | p. 21 |
| Rational Numbers | p. 28 |
| Review Exercises | p. 33 |
| Basic Notions of Groups | p. 35 |
| Definitions and Examples | p. 36 |
| Basic Properties | p. 41 |
| Subgroups | p. 45 |
| Generating Sets | p. 48 |
| Review Exercises | p. 51 |
| Cyclic Groups | p. 53 |
| Cyclic Groups | p. 54 |
| Subgroups of Cyclic Groups | p. 57 |
| Review Exercises | p. 63 |
| Permutation Groups | p. 65 |
| Symmetric Groups | p. 66 |
| Dihedral Groups | p. 71 |
| Alternating Groups | p. 76 |
| Review Exercises | p. 79 |
| Counting Theorems | p. 81 |
| Lagrange's Theorem | p. 82 |
| Conjugacy Classes of a Group | p. 87 |
| Review Exercises | p. 93 |
| Group Homomorphisms | p. 95 |
| Examples and Basic Properties | p. 96 |
| Isomorphisms | p. 99 |
| Cayley's Theorem | p. 105 |
| Review Exercises | p. 108 |
| The Quotient Group | p. 109 |
| Normal Subgroups | p. 110 |
| Quotient Groups | p. 114 |
| Fundamental Theorem of Group Homomorphisms | p. 119 |
| Review Exercises | p. 125 |
| Finite Abelian Groups | p. 127 |
| Direct Products of Groups | p. 128 |
| Cauchy's Theorem | p. 133 |
| Structure Theorem of Finite Abelian Groups | p. 137 |
| Review Exercises | p. 142 |
| Sylow Theorems and Applications | p. 143 |
| Group Actions | p. 144 |
| Sylow Theorems | p. 151 |
| Review Exercises | p. 157 |
| Introduction to Group Presentations | p. 159 |
| Free Groups and Free Abelian Groups | p. 160 |
| Generators and Relations | p. 165 |
| Classification of Finite Groups of Small Orders | p. 170 |
| Review Exercises | p. 175 |
| Types of Rings | p. 177 |
| Definitions and Examples | p. 178 |
| Matrix Rings | p. 185 |
| Review Exercises | p. 191 |
| Ideals and Quotient Rings | p. 193 |
| Ideals | p. 194 |
| Quotient Rings | p. 198 |
| Review Exercises | p. 203 |
| Ring Homomorphisms | p. 205 |
| Ring Homomorphisms | p. 206 |
| Direct Products of Rings | p. 211 |
| The Quotient Field of an Integral Domain | p. 216 |
| Review Exercises | p. 222 |
| Polynomial Rings | p. 223 |
| Polynomial Rings in the Indeterminates | p. 224 |
| Properties of the Polynomial Rings of One Variable | p. 228 |
| Principal Ideal Domains and Euclidean Domains | p. 233 |
| Review Exercises | p. 237 |
| Factorization | p. 239 |
| Irreducible and Prime Elements | p. 240 |
| Unique Factorization Domains | p. 245 |
| Polynomial Extensions of Factorial Domains | p. 253 |
| Review Exercises | p. 259 |
| Vector Spaces Over an Arbitrary Field | p. 261 |
| A Brief Review on Vector Spaces | p. 262 |
| A Brief Review on Linear Transformations | p. 266 |
| Review Exercises | p. 272 |
| Field Extensions | p. 273 |
| Algebraic or Transcendental? | p. 274 |
| Finite and Algebraic Extensions | p. 278 |
| Construction with Straightedge and Compass | p. 284 |
| Review Exercises | p. 294 |
| All About Roots | p. 295 |
| Zeros of Polynomials | p. 296 |
| Uniqueness of Splitting Fields | p. 299 |
| Algebraically Closed Fields | p. 303 |
| Multiplicity of Roots | p. 305 |
| Finite Fields | p. 309 |
| Review Exercises | p. 314 |
| Galois Pairing | p. 315 |
| Galois Groups | p. 316 |
| The Fixed Subfields of a Galois Group | p. 321 |
| Fundamental Theorem of Galois Pairing | p. 326 |
| Review Exercises | p. 331 |
| Applications of the Galois Pairing | p. 333 |
| Fields of Invariants | p. 334 |
| Solvable Groups | p. 338 |
| Insolvability of the Quintic | p. 345 |
| Review Exercises | p. 350 |
| Index | p. 351 |
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