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 | |
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