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| Preface | |
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| Preliminaries | |
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| Basic Ideas of Set Theory | |
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| Functions | |
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| Equivalence Relations and Partitions | |
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| A Note on Natural Numbers | |
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| Review Exercises | |
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| Algebraic Structure of Numbers | |
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| The Set of Integers | |
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| Congruences of Integers | |
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| Rational Numbers | |
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| Review Exercises | |
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| Basic Notions of Groups | |
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| Definitions and Examples | |
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| Basic Properties | |
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| Subgroups | |
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| Generating Sets | |
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| Review Exercises | |
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| Cyclic Groups | |
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| Cyclic Groups | |
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| Subgroups of Cyclic Groups | |
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| Review Exercises | |
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| Permutation Groups | |
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| Symmetric Groups | |
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| Dihedral Groups | |
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| Alternating Groups | |
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| Review Exercises | |
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| Counting Theorems | |
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| Lagrange's Theorem | |
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| Conjugacy Classes of a Group | |
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| Review Exercises | |
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| Group Homomorphisms | |
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| Examples and Basic Properties | |
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| Isomorphisms | |
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| Cayley's Theorem | |
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| Review Exercises | |
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| The Quotient Group | |
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| Normal Subgroups | |
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| Quotient Groups | |
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| Fundamental Theorem of Group Homomorphisms | |
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| Review Exercises | |
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| Finite Abelian Groups | |
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| Direct Products of Groups | |
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| Cauchy's Theorem | |
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| Structure Theorem of Finite Abelian Groups | |
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| Review Exercises | |
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| Sylow Theorems and Applications | |
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| Group Actions | |
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| Sylow Theorems | |
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| Review Exercises | |
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| Introduction to Group Presentations | |
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| Free Groups and Free Abelian Groups | |
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| Generators and Relations | |
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| Classification of Finite Groups of Small Orders | |
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| Review Exercises | |
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| Types of Rings | |
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| Definitions and Examples | |
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| Matrix Rings | |
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| Review Exercises | |
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| Ideals and Quotient Rings | |
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| Ideals | |
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| Quotient Rings | |
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| Review Exercises | |
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| Ring Homomorphisms | |
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| Ring Homomorphisms | |
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| Direct Products of Rings | |
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| The Quotient Field of an Integral Domain | |
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| Review Exercises | |
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| Polynomial Rings | |
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| Polynomial Rings in the Indeterminates | |
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| Properties of the Polynomial Rings of One Variable | |
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| Principal Ideal Domains and Euclidean Domains | |
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| Review Exercises | |
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| Factorization | |
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| Irreducible and Prime Elements | |
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| Unique Factorization Domains | |
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| Polynomial Extensions of Factorial Domains | |
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| Review Exercises | |
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| Vector Spaces Over an Arbitrary Field | |
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| A Brief Review on Vector Spaces | |
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| A Brief Review on Linear Transformations | |
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| Review Exercises | |
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| Field Extensions | |
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| Algebraic or Transcendental? | |
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| Finite and Algebraic Extensions | |
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| Construction with Straightedge and Compass | |
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| Review Exercises | |
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| All About Roots | |
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| Zeros of Polynomials | |
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| Uniqueness of Splitting Fields | |
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| Algebraically Closed Fields | |
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| Multiplicity of Roots | |
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| Finite Fields | |
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| Review Exercises | |
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| Galois Pairing | |
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| Galois Groups | |
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| The Fixed Subfields of a Galois Group | |
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| Fundamental Theorem of Galois Pairing | |
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| Review Exercises | |
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| Applications of the Galois Pairing | |
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| Fields of Invariants | |
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| Solvable Groups | |
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| Insolvability of the Quintic | |
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| Review Exercises | |
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| Index | |