Galois Theory

'1 Historical Aspects of the Resolution of Algebraic Equations.- 1.1 Approximating the Roots of an Equation.- 1.2 Construction of Solutions by Intersections of Curves.- 1.3 Relations with Trigonometry.- 1.4 Problems of Notation and Terminology.- 1.5 The Problem of Localization of the Roots.- 1.6 The Problem of the Existence of Roots.- 1.7 The Problem of Algebraic Solutions of Equations.- Toward Chapter 2.- 2 Resolution of Quadratic, Cubic, and Quartic Equations.- 2.1 Second-Degree Equations.- 2.1.1 The Babylonians.- 2.1.2 The Greeks.- 2.1.3 The Arabs.- 2.1.4 Use of Negative Numbers.- 2.2 Cubic Equations.- 2.2.1 The Greeks.- 2.2.2 Omar Khayyam and Sharaf ad Din at Tusi.- 2.2.3 Scipio del Ferro, Tartaglia, Cardan.- 2.2.4 Algebraic Solution of the Cubic Equation.- 2.2.5 First Computations with Complex Numbers.- 2.2.6 Raffaele Bombelli.- 2.2.7 François Viète.- 2.3 Quartic Equations.- Exercises for Chapter 2.- Solutions to Some of the Exercises.- 3 Symmetric Polynomials.- 3.1 Symmetric Minimal Polynomial of an Algebraic Element.- 4.4.4 Definition.- 4.4.5 Properties of the Minimal Polynomial.- 4.4.6 Proving the Irreducibility of a Polynomial in Z[X].- 4.5 Algebraic Extensions.- 4.5.1 Extensions Generated by an Algebraic Element.- 4.5.2 Properties of K[a].- 4.5.3 Definition.- 4.5.4 Extensions of Finite Degree.- 4.5.5 Corollary: Towers of Algebraic Extensions.- 4.6 Algebraic Extensions Generated by n Elements.- 4.6.1 Notation.- 4.6.2 Proposition.- 4.6.3 Corollary.- 4.7 Construction of an Extension by Adjoining a Root.- 4.7.1 Definition.- 4.7.2 Proposition.- 4.7.3 Corollary.- 4.7.4 Universal Property of K[X]/(P).- Toward Chapters 5 and 6.- Exercises for Chapter 4.- Solutions to Some of the Exercises.- 5 Constructions with Straightedge and Compass.- 5.1 Constructible Points.- 5.2 Examples of Classical Constructions.- 5.2.1 Projection of a Point onto a Line.- 5.2.2 Construction of an Orthonormal Basis from Two Points.- 5.2.3 Construction of a Line Parallel to a Given Line

J.-P. Escofier