| |
| |
| Introduction | |
| |
| |
| |
| Mappings and Operations | |
| |
| |
| |
| Mappings | |
| |
| |
| |
| Composition. Invertible Mappings | |
| |
| |
| |
| Operations | |
| |
| |
| |
| Composition as an Operation | |
| |
| |
| |
| Introduction to Groups | |
| |
| |
| |
| Definition and Examples | |
| |
| |
| |
| Permutations | |
| |
| |
| |
| Subgroups | |
| |
| |
| |
| Groups and Symmetry | |
| |
| |
| |
| Equivalence. Congruence. Divisibility | |
| |
| |
| |
| Equivalence Relations | |
| |
| |
| |
| Congruence. The Division Algorithm | |
| |
| |
| |
| Integers Modulo n | |
| |
| |
| |
| Greatest Common Divisors. The Euclidean Algorithm | |
| |
| |
| |
| Factorization. Euler's Phi-Function | |
| |
| |
| |
| Groups | |
| |
| |
| |
| Elementary Properties | |
| |
| |
| |
| Generators. Direct Products | |
| |
| |
| |
| Cosets | |
| |
| |
| |
| Lagrange's Theorem. Cyclic Groups | |
| |
| |
| |
| Isomorphism | |
| |
| |
| |
| More on Isomorphism | |
| |
| |
| |
| Cayley's Theorem | |
| |
| |
| |
| RSA Algorithm | |
| |
| |
| |
| Group Homomorphisms | |
| |
| |
| |
| Homomorphisms of Groups. Kernels | |
| |
| |
| |
| Quotient Groups | |
| |
| |
| |
| The Fundamental Homomorphism Theorem | |
| |
| |
| |
| Introduction to Rings | |
| |
| |
| |
| Definition and Examples | |
| |
| |
| |
| Integral Domains. Subrings | |
| |
| |
| |
| Fields | |
| |
| |
| |
| Isomorphism. Characteristic | |
| |
| |
| |
| The Familiar Number Systems | |
| |
| |
| |
| Ordered Integral Domains | |
| |
| |
| |
| The Integers | |
| |
| |
| |
| Field of Quotients. The Field of Rational Numbers | |
| |
| |
| |
| Ordered Fields. The Field of Real Numbers | |
| |
| |
| |
| The Field of Complex Numbers | |
| |
| |
| |
| Complex Roots of Unity | |
| |
| |
| |
| Polynomials | |
| |
| |
| |
| Definition and Elementary Properties | |
| |
| |
| Appendix to Section 34 | |
| |
| |
| |
| The Division Algorithm | |
| |
| |
| |
| Factorization of Polynomials | |
| |
| |
| |
| Unique Factorization Domains | |
| |
| |
| |
| Quotient Rings | |
| |
| |
| |
| Homomorphisms of Rings. Ideals | |
| |
| |
| |
| Quotient Rings | |
| |
| |
| |
| Quotient Rings of F[X] | |
| |
| |
| |
| Factorization and Ideals | |
| |
| |
| |
| Galois Theory: Overview | |
| |
| |
| |
| Simple Extensions. Degree | |
| |
| |
| |
| Roots of Polynomials | |
| |
| |
| |
| Fundamental Theorem: Introduction | |
| |
| |
| |
| Galois Theory | |
| |
| |
| |
| Algebraic Extensions | |
| |
| |
| |
| Splitting Fields. Galois Groups | |
| |
| |
| |
| Separability and Normality | |
| |
| |
| |
| Fundamental Theorem of Galois Theory | |
| |
| |
| |
| Solvability by Radicals | |
| |
| |
| |
| Finite Fields | |
| |
| |
| |
| Geometric Constructions | |
| |
| |
| |
| Three Famous Problems | |
| |
| |
| |
| Constructible Numbers | |
| |
| |
| |
| Impossible Constructions | |
| |
| |
| |
| Solvable and Alternating Groups | |
| |
| |
| |
| Isomorphism Theorems and Solvable Groups | |
| |
| |
| |
| Alternating Groups | |
| |
| |
| |
| Applications of Permutation Groups | |
| |
| |
| |
| Groups Acting on Sets | |
| |
| |
| |
| Burnside's Counting Theorem | |
| |
| |
| |
| Sylow's Theorem | |
| |
| |
| |
| Symmetry | |
| |
| |
| |
| Finite Symmetry Groups | |
| |
| |
| |
| Infinite Two-Dimensional Symmetry Groups | |
| |
| |
| |
| On Crystallographic Groups | |
| |
| |
| |
| The Euclidean Group | |
| |
| |
| |
| Lattices and Boolean Algebras | |
| |
| |
| |
| Partially Ordered Sets | |
| |
| |
| |
| Lattices | |
| |
| |
| |
| Boolean Algebras | |
| |
| |
| |
| Finite Boolean Algebras | |
| |
| |
| |
| Sets | |
| |
| |
| |
| Proofs | |
| |
| |
| |
| Mathematical Induction | |
| |
| |
| |
| Linear Algebra | |
| |
| |
| |
| Solutions to Selected Problems | |
| |
| |
| Photo Credit List | |
| |
| |
| Index of Notation | |
| |
| |
| Index | |