| Acknowledgments | |
| Galois | |
| Influence of Lagrange | |
| Quadratic equations | |
| 1700 B.C. to A.D. 1500 | |
| Solution of cubic | |
| Solution of quartic | |
| Impossibility of quintic | |
| Newton | |
| Symmetric polynomials in roots | |
| Fundamental theorem on symmetric polynomials | |
| Proof | |
| Newton's theorem | |
| Discriminants | |
| Solution of cubic | |
| Lagrange and Vandermonde | |
| Lagrange resolvents | |
| Solution of quartic again | |
| Attempt at quintic | |
| Lagrange's Reflexions | |
| Cyclotomic equations | |
| The cases n = 3, 5 | |
| n = 7, 11 | |
| General case | |
| Two lemmas | |
| Gauss's method | |
| p-gons by ruler and compass | |
| Summary | |
| Resolvents | |
| Lagrange's theorem | |
| Proof | |
| Galois resolvents | |
| Existence of Galois resolvents | |
| Representation of the splitting field as K(t) | |
| Simple algebraic extensions | |
| Euclidean algorithm | |
| Construction of simple algebraic extensions | |
| Galois' method | |
| Review | |
| Finite permutation groups | |
| Subgroups, normal subgroups | |
| The Galois group of an equation | |
| Examples | |
| Solvability by radicals | |
| Reduction of the Galois group by a cyclic extension | |
| Solvable groups | |
| Reduction to a normal subgroup of index p | |
| Theorem on solution by radicals (assuming roots of unity) | |
| Summary | |
| Splitting fields | |
| Fundamental theorem of algebra (so-called) | |
| Construction of a splitting field | |
| Need for a factorization method | |
| Three theorems on factorization methods | |
| Uniqueness of factorization of polynomials | |
| Factorization over Z | |
| Over Q | |
| Gauss's lemma, factorization over Q | |
| Over transcendental extensions | |
| Of polynomials in two variables | |
| Over algebraic extensions | |
| Final remarks | |
| Review of Galois theory | |
| Fundamental theorem of Galois theory (so-called) | |
| Galois group of x[superscript p] - 1 = 0 over Q | |
| Solvability of the cyclotomic equation | |
| Theorem on solution by radicals | |
| Equations with literal coefficients | |
| Equations of prime degree | |
| Galois group of x[superscript n] - 1 = 0 over Q | |
| Proof of the main proposition | |
| Deduction of Lemma 2 of 24 | |
| Memoir on the Conditions for Solvability of Equations by Radicals, by Evariste Galois | |
| Synopsis | |
| Groups | |
| Answers to Exercises | |
| List of Exercises | |
| References | |
| Index | |
| Table of Contents provided by Blackwell. All Rights Reserved. |