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