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| Preface | |
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| Acknowledgments | |
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| Background | |
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| Sets and Maps | |
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| Equivalence Relations and Partitions | |
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| Properties of Z | |
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| Complex Numbers | |
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| Matrices | |
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| Group Theory | |
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| Groups | |
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| Examples and Basic Concepts | |
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| Subgroups | |
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| Cyclic Groups | |
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| Permutations | |
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| Group Homomorphisms | |
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| Cosets and Lagrange's Theorem | |
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| Homomorphisms | |
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| Normal Subgroups | |
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| Quotient Groups | |
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| Automorphisms | |
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| Direct Products and Abelian Groups | |
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| Examples and Definitions | |
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| Computing Orders | |
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| Direct Sums | |
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| Fundamental Theorem of Finite Abelian Groups | |
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| Group Actions | |
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| Group Actions and Cayley's Theorem | |
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| Stabilizers and Orbits in a Group Action | |
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| Burnside's Theorem and Applications | |
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| Conjugacy Classes and the Class Equation | |
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| Conjugacy in S[subscript n] and Simplicity of A[subscript 5] | |
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| The Sylow Theorems | |
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| Applications of the Sylow Theorems | |
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| Composition Series | |
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| Isomorphism Theorems | |
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| The Jordan-Holder Theorem | |
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| Solvable Groups | |
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| Rings and Fields | |
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| Rings | |
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| Examples and Basic Concepts | |
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| Integral Domains | |
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| Fields | |
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| Ring Homomorphisms | |
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| Definitions and Basic Properties | |
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| Ideals | |
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| The Field of Quotients | |
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| Rings of Polynomials | |
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| Basic Concepts and Notation | |
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| The Division Algorithm in F[x] | |
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| More Applications of the Division Algorithm | |
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| Irreducible Polynomials | |
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| Cubic and Quartic Polynomials | |
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| Ideals in F[x] | |
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| Quotient Rings of F[x] | |
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| The Chinese Remainder Theorem for F[x] | |
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| Euclidean Domains | |
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| Division Algorithms and Euclidean Domains | |
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| Unique Factorization Domains | |
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| Gaussian Integers | |
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| Field Theory | |
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| Vector Spaces | |
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| Algebraic Extensions | |
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| Splitting Fields | |
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| Finite Fields | |
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| Geometric Constructions | |
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| Constructible Real Numbers | |
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| Classical Problems | |
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| Constructions with Marked Ruler and Compass | |
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| Cubics and Quartics Revisited | |
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| Galois Theory | |
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| Galois Groups | |
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| The Fundamental Theorem of Galois Theory | |
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| Galois Groups of Polynomials | |
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| Geometric Constructions Revisited | |
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| Radical Extensions | |
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| Historical Notes | |
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| From Ahmes the Scribe to Omar Khayyam | |
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| From Gerolamo Cardano to C. F. Gauss | |
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| From Evariste Galois to Emmy Noether | |
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| Selected Topics | |
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| Symmetries | |
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| Linear Transformations | |
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| Isometries | |
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| Symmetry Groups | |
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| Platonic Solids | |
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| Subgroups of the Special Orthogonal Group | |
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| Further Reading | |
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| Grobner Bases | |
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| Lexicographic Order | |
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| A Division Algorithm | |
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| Dickson's Lemma | |
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| The Hilbert Basis Theorem | |
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| Grobner Bases and the Division Algorithm | |
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| Further Reading | |
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| Coding Theory | |
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| Linear Binary Codes | |
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| Error Correction and Coset Decoding | |
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| Standard Generator Matrices | |
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| The Syndrome Method | |
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| Cyclic Codes | |
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| Further Reading | |
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| Boolean Algebras | |
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| Lattices | |
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| Boolean Algebras | |
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| Circuits | |
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| Further Reading | |
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| Answers and Hints to Selected Exercises | |
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| Bibliography | |
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| Index | |