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Introduction | p. 1 |
Mappings and Operations | p. 9 |
Mappings | p. 9 |
Composition. Invertible Mappings | p. 15 |
Operations | p. 19 |
Composition as an Operation | p. 25 |
Introduction to Groups | p. 30 |
Definition and Examples | p. 30 |
Permutations | p. 34 |
Subgroups | p. 41 |
Groups and Symmetry | p. 47 |
Equivalence. Congruence. Divisibility | p. 52 |
Equivalence Relations | p. 52 |
Congruence. The Division Algorithm | p. 57 |
Integers Modulo n | p. 61 |
Greatest Common Divisors. The Euclidean Algorithm | p. 65 |
Factorization. Euler's Phi-Function | p. 70 |
Groups | p. 75 |
Elementary Properties | p. 75 |
Generators. Direct Products | p. 81 |
Cosets | p. 85 |
Lagrange's Theorem. Cyclic Groups | p. 88 |
Isomorphism | p. 93 |
More on Isomorphism | p. 98 |
Cayley's Theorem | p. 102 |
RSA Algorithm | p. 105 |
Group Homomorphisms | p. 106 |
Homomorphisms of Groups. Kernels | p. 106 |
Quotient Groups | p. 110 |
The Fundamental Homomorphism Theorem | p. 114 |
Introduction to Rings | p. 120 |
Definition and Examples | p. 120 |
Integral Domains. Subrings | p. 125 |
Fields | p. 128 |
Isomorphism. Characteristic | p. 131 |
The Familiar Number Systems | p. 137 |
Ordered Integral Domains | p. 137 |
The Integers | p. 140 |
Field of Quotients. The Field of Rational Numbers | p. 142 |
Ordered Fields. The Field of Real Numbers | p. 146 |
The Field of Complex Numbers | p. 149 |
Complex Roots of Unity | p. 154 |
Polynomials | p. 160 |
Definition and Elementary Properties | p. 160 |
Appendix to Section 34 | p. 162 |
The Division Algorithm | p. 165 |
Factorization of Polynomials | p. 169 |
Unique Factorization Domains | p. 173 |
Quotient Rings | p. 178 |
Homomorphisms of Rings. Ideals | p. 178 |
Quotient Rings | p. 182 |
Quotient Rings of F[X] | p. 184 |
Factorization and Ideals | p. 187 |
Galois Theory: Overview | p. 193 |
Simple Extensions. Degree | p. 194 |
Roots of Polynomials | p. 198 |
Fundamental Theorem: Introduction | p. 203 |
Galois Theory | p. 207 |
Algebraic Extensions | p. 207 |
Splitting Fields. Galois Groups | p. 210 |
Separability and Normality | p. 214 |
Fundamental Theorem of Galois Theory | p. 218 |
Solvability by Radicals | p. 219 |
Finite Fields | p. 223 |
Geometric Constructions | p. 229 |
Three Famous Problems | p. 229 |
Constructible Numbers | p. 233 |
Impossible Constructions | p. 234 |
Solvable and Alternating Groups | p. 237 |
Isomorphism Theorems and Solvable Groups | p. 237 |
Alternating Groups | p. 240 |
Applications of Permutation Groups | p. 243 |
Groups Acting on Sets | p. 243 |
Burnside's Counting Theorem | p. 247 |
Sylow's Theorem | p. 252 |
Symmetry | p. 256 |
Finite Symmetry Groups | p. 256 |
Infinite Two-Dimensional Symmetry Groups | p. 263 |
On Crystallographic Groups | p. 267 |
The Euclidean Group | p. 274 |
Lattices and Boolean Algebras | p. 279 |
Partially Ordered Sets | p. 279 |
Lattices | p. 283 |
Boolean Algebras | p. 287 |
Finite Boolean Algebras | p. 291 |
Sets | p. 296 |
Proofs | p. 299 |
Mathematical Induction | p. 304 |
Linear Algebra | p. 307 |
Solutions to Selected Problems | p. 312 |
Photo Credit List | p. 326 |
Index of Notation | p. 327 |
Index | p. 330 |
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