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| Preface to the First Edition | |
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| Preface to the Second Edition | |
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| Notation | |
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| Basic Notation | |
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| Chapter-by-Chapter Notation | |
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| Polynomials | |
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| Cubic Equations | |
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| Cardan's Formulas | |
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| Permutations of the Roots | |
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| Cubic Equations over the Real Numbers | |
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| Symmetric Polynomials | |
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| Polynomials of Several Variables | |
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| Symmetric Polynomials | |
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| Computing with Symmetric Polynomials (Optional) | |
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| The Discriminant | |
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| Roots of Polynomials | |
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| The Existence of Roots | |
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| The Fundamental Theorem of Algebra | |
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| Fields | |
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| Extension Fields | |
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| Elements of Extension Fields | |
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| Irreducible Polynomials | |
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| The Degree of an Extension | |
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| Algebraic Extensions | |
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| Normal and Separable Extensions | |
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| Splitting Fields | |
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| Normal Extensions | |
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| Separable Extensions | |
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| Theorem of the Primitive Element | |
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| The Galois Group | |
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| Definition of the Galois Group | |
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| Galois Groups of Splitting Fields | |
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| Permutations of the Roots | |
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| Examples of Galois Groups | |
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| Abelian Equations (Optional) | |
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| The Galois Correspondence | |
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| Galois Extensions | |
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| Normal Subgroups and Normal Extensions | |
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| The Fundamental Theorem of Galois Theory | |
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| First Applications | |
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| Automorphisms and Geometry (Optional) | |
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| Applications | |
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| Solvability by Radicals | |
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| Solvable Groups | |
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| Radical and Solvable Extensions | |
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| Solvable Extensions and Solvable Groups | |
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| Simple Groups | |
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| Solving Polynomials by Radicals | |
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| The Casus Irreducbilis (Optional) | |
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| Cyclotomic Extensions | |
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| Cyclotomic Polynomials | |
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| Gauss and Roots of Unity (Optional) | |
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| Geometric Constructions | |
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| Constructible Numbers | |
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| Regular Polygons and Roots of Unity | |
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| Origami (Optional) | |
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| Finite Fields | |
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| The Structure of Finite Fields | |
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| Irreducible Polynomials over Finite Fields (Optional) | |
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| Further Topics | |
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| Lagrange, Galois, and Kronecker | |
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| Lagrange | |
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| Galois | |
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| Kronecker | |
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| Computing Galois Groups | |
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| Quartic Polynomials | |
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| Quintic Polynomials | |
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| Resolvents | |
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| Other Methods | |
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| Solvable Permutation Groups | |
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| Polynomials of Prime Degree | |
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| Imprimitive Polynomials of Prime-Squared Degree | |
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| Primitive Permutation Groups | |
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| Primitive Polynomials of Prime-Squared Degree | |
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| The Lemniscate | |
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| Division Points and Arc Length | |
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| The Lemniscatic Function | |
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| The Complex Lemniscatic Function | |
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| Complex Multiplication | |
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| Abel's Theorem | |
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| Abstract Algebra | |
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| Basic Algebra | |
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| Complex Numbers | |
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| Polynomials with Rational Coefficients | |
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| Group Actions | |
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| More Algebra | |
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| Index | |