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