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'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 Francois Viete.- 2.3 Quartic Equations.- Exercises for Chapter 2.- Solutions to Some of the Exercises.- 3 Symmetric Polynomials.- 3.1 Symmetric Polynomials.- 3.1.1 Background.- 3.1.2 Definitions.- 3.2 Elementary Symmetric Polynomials.- 3.2.1 Definition.- 3.2.2 The Product of the X ? Xi; Relations Between Coefficients and Roots.- 3.3 Symmetric Polynomials and Elementary Symmetric Polynomials.- 3.3.1 Theorem.- 3.3.2 Proposition.- 3.3.3 Proposition.- 3.4 Newton's Formulas.- 3.5 Resultant of Two Polynomials.- 3.5.1 Definition.- 3.5.2 Proposition.- 3.6 Discriminant of a Polynomial.- 3.6.1 Definition.- 3.6.2 Proposition.- 3.6.3 Formulas.- 3.6.4 Polynomials with Real Coefficients: Real Roots and Sign of the Discriminant.- Exercises for Chapter 3.- Solutions to Some of the Exercises.- 4 Field Extensions.- 4.1 Field Extensions.- 4.1.1 Definition.- 4.1.2 Proposition.- 4.1.3 The Degree of an Extension.- 4.1.4 Towers of Fields.- 4.2 The Tower Rule.- 4.2.1 Proposition.- 4.3 Generated Extensions.- 4.3.1 Proposition.- 4.3.2 Definition.- 4.3.3 Proposition.- 4.4 Algebraic Elements.- 4.4.1 Definition.- 4.4.2 Transcendental Numbers.- 4.4.3 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 Passing Through a Point.- 5.3 Lemma.- 5.4 Coordinates of Points Constructible in One Step.- 5.5 A Necessary Condition for Constructibility.- 5.6 Two Problems More Than Two Thousand Years Old.- 5.6.1 Duplication of the Cube.- 5.6.2 Trisection of the Angle.- 5.7 A Sufficient Condition for Constructibility.- Exercises for Chapter 5.- Solutions to Some of the Exercises.- 6 K-Homomorphisms.- 6.1 Conjugate Numbers.- 6.2 K-Homomorphisms.- 6.2.1 Definitions.- 6.2.2 Properties.- 6.3 Algebraic Elements and K-Homomorphisms.- 6.3.1 Proposition.- 6.3.2 Example.- 6.4 Extensions of Embeddings into ?.- 6.4.1 Definition.- 6.4.2 Proposition.- 6.4.3 Proposition.- 6.5 The Primitive Element Theorem.- 6.5.1 Theorem and Definition.- 6.5.2 Example.- 6.6 Linear Independence of K-Homomorphisms.- 6.6.1 Characters.- 6.6.2 Emil Artin's Theorem.- 6.6.3 Corollary: Dedekind's Theorem.- Exercises for Chapter 6.- Solutions to Some

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