| |
| |
| Introduction | |
| |
| |
| Historical Note: Al-Khwarizmi | |
| |
| |
| |
| Preliminaries | |
| |
| |
| |
| Properties of the Integers | |
| |
| |
| Historical Note: Augustus de Morgan | |
| |
| |
| |
| Solving Cubic and Quartic Polynomial Equations | |
| |
| |
| Historical Note: How the Cubic and Quartic Equations Were Solved | |
| |
| |
| |
| Complex Numbers | |
| |
| |
| Historical Note: Highlights in the Development of the Complex Numbers | |
| |
| |
| |
| Some Other Examples | |
| |
| |
| Historical Note: William Rowan Hamilton | |
| |
| |
| |
| Algebraic Extension Fields | |
| |
| |
| |
| Fields | |
| |
| |
| |
| Solvability by Radicals | |
| |
| |
| Historical Note: Niels Henrik Abel | |
| |
| |
| |
| Rings | |
| |
| |
| |
| Ways in Which Polynomials Are Like the Integers | |
| |
| |
| Historical Note: Julia Robinson | |
| |
| |
| |
| Principal Ideals | |
| |
| |
| Historical Note: Emmy Noether | |
| |
| |
| |
| Algebraic Elements | |
| |
| |
| |
| Eisenstein's Irreducibility Criterion | |
| |
| |
| Historical Note: Gotthold Eisenstein | |
| |
| |
| |
| Extension Fields as Vector Spaces | |
| |
| |
| |
| Automorphisms to Fields | |
| |
| |
| Historical Note: Evariste Galois | |
| |
| |
| |
| Counting Automorphisms | |
| |
| |
| Historical Note: Richard Dedekind | |
| |
| |
| |
| Elementary Group Theory | |
| |
| |
| |
| Groups | |
| |
| |
| Historical Note: Walther von Dyck | |
| |
| |
| |
| Permutation Groups | |
| |
| |
| |
| Group Homomorphisms | |
| |
| |
| Historical Note: Arthur Cayley | |
| |
| |
| |
| Subgroups | |
| |
| |
| |
| Subgroups Generated by Subsets | |
| |
| |
| |
| Cosets | |
| |
| |
| |
| Finite Groups and Lagrange's Theorem | |
| |
| |
| Historical Note: Joseph Louis Lagrange | |
| |
| |
| |
| Equivalence Relations and Cauchy's Theorem | |
| |
| |
| Historical Note: Augustin-Louis Cauchy | |
| |
| |
| |
| Normal Subgroups and Quotient Groups | |
| |
| |
| Historical Note: Otto Holder | |
| |
| |
| |
| The Homomorphism Theorem for Groups | |
| |
| |
| Historical Note: B. L. van der Waerden | |
| |
| |
| |
| Polynomial Equations Not Solvable by Radicals | |
| |
| |
| |
| Galois Groups of Radical Extensions | |
| |
| |
| |
| Solvable Groups and Commutator Subgroups | |
| |
| |
| Historical Note: William Burnside | |
| |
| |
| |
| Solvable Galois Groups | |
| |
| |
| |
| Polynomial Equations Not Solvable by Radicals | |
| |
| |
| Historical Note: Paolo Ruffini | |
| |
| |
| |
| Finite Groups | |
| |
| |
| |
| Finite External Direct Products of Groups | |
| |
| |
| Historical Note: J. H. M. Wedderburn | |
| |
| |
| |
| Finite Internal Direct Products of Groups | |
| |
| |
| |
| Abelian Groups with Prime Power Order | |
| |
| |
| |
| The Fundamental Theorem of Finite Abelian Groups | |
| |
| |
| Historical Note: Leopold Kronecker | |
| |
| |
| |
| Dihedral Groups | |
| |
| |
| Historical Note: Felix Klein | |
| |
| |
| |
| Cauchy's Theorem | |
| |
| |
| |
| The Sylow Theorems | |
| |
| |
| Historical Note: Peter Ludvig Sylow | |
| |
| |
| |
| Groups of Order Less Than 16 | |
| |
| |
| |
| Groups of Even Permutations | |
| |
| |
| Historical Note: Camille Jordan | |
| |
| |
| |
| Semidirect Products | |
| |
| |
| |
| The Greek Alphabet | |
| |
| |
| |
| Proving Theorems | |
| |
| |
| Historical Note: George Boole | |
| |
| |
| |
| Vector Spaces Over Fields | |
| |
| |
| |
| Constructions with Straightedge and Compass | |
| |
| |
| Answers to Odd-Numbered Computational Exercises | |
| |
| |
| Bibliography | |
| |
| |
| Photo Credits | |
| |
| |
| Notation Index | |
| |
| |
| Subject Index | |