Jean-Pierre Escofier. Galois Theory. Translated by Leila Schneps. With 48 Illustrations. Springer

Transcription

1 Jean-Pierre Escofier Galois Theory Translated by Leila Schneps With 48 Illustrations Springer

2 Preface v 1 Historical Aspects of the Resolution of Algebraic Equations Approximating the Roots of an Equation Construction of Solutions by Intersections of Curves Relations with Trigonometry Problems of Notation and Terminology The Problem of Localization of the Roots The Problem of the Existence of Roots The Problem of Algebraic Solutions of Equations 6 '" Toward Chapter Resolution of Quadratic, Cubic, and Quartic Equations Second-Degree Equations The Babylonians The Greeks The Arabs Use of Negative Numbers Cubic Equations The Greeks Omar Khayyam and Sharaf ad Din at Tusi Scipio del Ferro, Tartaglia, Cardan Algebraic Solution of the Cubic Equation First Computations with Complex Numbers Raifaele Bombelli 17

3 viii Contents Francois Viete Quartic Equations 18 Exercises for Chapter 2 19 Solutions to Some of the Exercises 22 3 Symmetric Polynomials Symmetric Polynomials Background Definitions Elementary Symmetric Polynomials Definition The Product of the X - X*; Relations Between Coefficients and Roots Symmetric Polynomials and Elementary Symmetric Polynomials Theorem Proposition Proposition Newton's Formulas Resultant of Two Polynomials Definition Proposition Discriminant of a Polynomial Definition Proposition Formulas Polynomials with Real Coefficients: Real Roots and Sign of the Discriminant 38 Exercises for Chapter 3 39 Solutions to Some of the Exercises 44 4 Field Extensions Field Extensions Definition Proposition The Degree of an Extension Towers of Fields The Tower Rule Proposition Generated Extensions Proposition Definition" Proposition Algebraic Elements Definition 55

4 ix Transcendental Numbers Minimal Polynomial of an Algebraic Element Definition _ Properties of the Minimal Polynomial Proving the Irreducibility of a Polynomial in Z[X] Algebraic Extensions Extensions Generated by an Algebraic Element Properties of if [a] Definition Extensions of Finite Degree Corollary: Towers of Algebraic Extensions Algebraic Extensions Generated by n Elements Notation Proposition Corollary Construction of an Extension by Adjoining a Root Definition Proposition Corollary Universal Property of K[X]/(P) 63 Toward Chapters 5 and 6 64 Exercises for Chapter 4 64 Solutions to Some of the Exercises 69 Constructions with Straightedge and Compass Constructible Points Examples of Classical Constructions Projection of a Point onto a Line Construction of an Orthonormal Basis from Two Points Construction of a Line Parallel to a Given Line Pass- > ing Through a Point Lemma Coordinates of Points Constructible in One Step A Necessary Condition for Constructibility [ Two Problems More Than Two Thousand Years Old Duplication of the Cube Trisection of the Angle A Sufficient Condition for Constructibility 85 Exercises for Chapter 5 87 Solutions to Some of the Exercises 90 if-homomorphisms ^ Conjugate Numbers if-homomorphisms Definitions 94

5 6.2.2 Properties Algebraic Elements and if-homomorphisms Proposition Example Extensions of Embeddings into C Definition Proposition Proposition The Primitive Element Theorem Theorem and Definition Example Linear Independence of if-homomorphisms Characters Emil Artin's Theorem Corollary: Dedekind's Theorem 102 Exercises for Chapter Solutions to Some of the Exercises " 103 Normal Extensions Splitting Fields Definition Splitting Field of a Cubic Polynomial Normal Extensions Normal Extensions and if-homomorphisms Splitting Fields and Normal Extensions Proposition Converse Normal Extensions and Intermediate Extensions Normal Closure Ill Definition Ill Proposition Ill Proposition Ill 7.7 Splitting Fields: General Case 112 Toward Chapter Exercises for Chapter Solutions to Some of the Exercises 115 Galois Groups Galois Groups The Galois Group of an Extension The Order of the Galois Group of a Normal Extension of-finite Degree The Galois Group of a Polynomial The Galois Group as a Subgroup of a Permutation Group 120

6 xi A Short History of Groups Fields of Invariants Definition and Proposition ^ Emil Artin's Theorem The^Example of Q [v^,j]: First Part Galois Groups and Intermediate Extensions The Galois Correspondence The Example of Q [\/2,j]: Second Part The Example X Dihedral Groups The Special Case of D A The Galois Group of X The Galois Correspondence Search for Minimal Polynomials 132 Toward Chapters 9, 10, and Exercises for Chapter Solutions to Some of the Exercises Roots of Unity The Group U(n) of Units of the Ring Z/nZ Definition and Background The Structure of U(n) The Mobius Function Multiplicative Functions The Mobius Function Proposition The Mobius Inversion Formula Roots of Unity n-th Roots of Unity Proposition 153._ Primitive Roots Properties of Primitive Roots Cyclotomic Polynomials Definition Properties of the Cyclotomic Polynomial The Galois Group over Q of an Extension of Q by a Root of Unity 156 Exercises for Chapter Solutions to Some of the Exercises Cyclic Extensions Cyclic and Abelian Extensions Extensions by a Root and Cyclic Extensions Irreducibility of X p - a Hilbert's Theorem

7 xii Contents The Norm Hilbert's Theorem Extensions by a Root and Cyclic Extensions: Converse Lagrange Resolvents Definition Properties Resolution of the Cubic Equation Solution of the Quartic Equation Historical Commentary 188 Exercises for Chapter Solutions to Some of the Exercises Solvable Groups First Definition Derived or Commutator Subgroup Second Definition of Solvability Examples of Solvable Groups Third Definition The Group A n Is Simple for n > Theorem An Is Not Solvable for n > 5, Direct Proof Recent Results." 199 Exercises for Chapter Solutions to Some of the Exercises Solvability of Equations by Radicals Radical Extensions and Polynomials Solvable by Radicals Radical Extensions Polynomials Solvable by Radicals First Construction Second Construction If a Polynomial Is Solvable by Radicals, Its Galois Group Is Solvable Example of a Polynomial Not Solvable by Radicals The Converse of the Fundamental Criterion The General Equation of Degree n Algebraically Independent Elements Existence of Algebraically Independent Elements The General Equation of Degree n Galois Group of the General Equation of Degree n. 211 Exercises for Chapter 12...' 212 Solutions to Some of the Exercises The Life of Evariste Galois 219

8 xiii 14 Finite Fields Algebraically Closed Fields Definition Algebraic Closures Theorem (Steinitz, 1910) Examples of Finite Fields The Characteristic of a Field Definition Properties Properties of Finite Fields Proposition The Frobenius Homomorphism Existence and Uniqueness of a Finite Field with p r Elements Proposition Corollary Extensions of Finite Fields Normality of a Finite Extension of Finite Fields The Galois Group of a Finite Extension of a Finite Field Proposition The Galois Correspondence Example 234 Exercises for Chapter Solutions to Some of the Exercises Separable Extensions Separability Example of an Inseparable Element A Criterion for Separability Perfect Fields Perfect Fields and Separable Extensions Galois Extensions Definition Proposition The Galois Correspondence 260 Toward Chapter 16 ; Recent Developments The Inverse Problem of Galois Theory The Problem The Abelian Case Example; Computation of Galois Groups over Q for Small-Degree Polynomials Simplification of the Problem The Irreducibility Problem 263

9 xiv Contents Embedding of G into S n Looking for G Among the Transitive Subgroups of S n Transitive Subgroups of Study of $(G) C A n Study of $(G) cd Study of $(G) C Z/4Z An Algorithm for n"= Bibliography 271 Index 277