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 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 of the Exercises.- 7 Normal Extensions.- 7.1 Splitting Fields.- 7.1.1 Definition.- 7.1.2 Splitting Field of a Cubic Polynomial.- 7.2 Normal Extensions.- 7.3 Normal Extensions and K-Homomorphisms.- 7.4 Splitting Fields and Normal Extensions.- 7.4.1 Proposition.- 7.4.2 Converse.- 7.5 Normal Extensions and Intermediate Extensions.- 7.6 Normal Closure.- 7.6.1 Definition.- 7.6.2 Proposition.- 7.6.3 Proposition.- 7.7 Splitting Fields: General Case.- Toward Chapter 8.- Exercises for Chapter 7.- Solutions to Some of the Exercises.- 8 Galois Groups.- 8.1 Galois Groups.- 8.1.1 The Galois Group of an Extension.- 8.1.2 The Order of the Galois Group of a Normal Extension of Finite Degree.- 8.1.3 The Galois Group of a Polynomial.- 8.1.4 The Galois Group as a Subgroup of a Permutation Group.- 8.1.5 A Short History of Groups.- 8.2 Fields of Invariants.- 8.2.1 Definition and Proposition.- 8.2.2 Emil Artin’s Theorem.- 8.3 The Example of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaacbaGae8xuae1aamWaaeaadaGcbaqaaiabikdaYaWc % baGaeG4mamdaaOGaeiilaWIaemOAaOgacaGLBbGaayzxaaaaaa!4235! $$ Qleft[ {sqrt[3]{2},j} right] $$ : First Part.- 8.4 Galois Groups and Intermediate Extensions.- 8.5 The Galois Correspondence.- 8.6 The Example of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaacbaGae8xuae1aamWaaeaadaGcbaqaaiabikdaYaWc % baGaeG4mamdaaOGaeiilaWIaemOAaOgacaGLBbGaayzxaaaaaa!4235! $$ Qleft[ {sqrt[3]{2},j} right] $$: Second Part.- 8.7 The Example X4 + 2.- 8.7.1 Dihedral Groups.- 8.7.2 The Special Case of D4.- 8.7.3 The Galois Group of X4 + 2.- 8.7.4 The Galois Correspondence.- 8.7.5 Search for Minimal Polynomials.- Toward Chapters 9, 10, and 12.- Exercises for Chapter 8.- Solutions to Some of the Exercises.- 9 Roots of Unity.- 9.1 The Group U(n) of Units of the Ring ?/n?.- 9.1.1 Definition and Background.- 9.1.2 The Structure of U(n).- 9.2 The Möbius Function.- 9.2.1 Multiplicative Functions.- 9.2.2 The Möbius Function.- 9.2.3 Proposition.- 9.2.4 The Möbius Inversion Formula.- 9.3 Roots of Unity.- 9.3.1 n-th Roots of Unity.- 9.3.2 Proposition.- 9.3.3 Primitive Roots.- 9.3.4 Properties of Primitive Roots.- 9.4 Cyclotomic Polynomials.- 9.4.1 Definition.- 9.4.2 Properties of the Cyclotomic Polynomial.- 9.5 The Galois Group over Q of an Extension of Q by a Root of Unity.- Exercises for Chapter 9.- Solutions to Some of the Exercises.- 10 Cyclic Extensions.- 10.1 Cyclic and Abelian Extensions.- 10.2 Extensions by a Root and Cyclic Extensions.- 10.3 Irreducibility of Xp ? a.- 10.4 Hilbert’s Theorem 90.- 10.4.1 The Norm.- 10.4.2 Hilbert’s Theorem 90.- 10.5 Extensions by a Root and Cyclic Extensions: Converse.- 10.6 Lagrange Resolvents.- 10.6.1 Definition.- 10.6.2 Properties.- 10.7 Resolution of the Cubic Equation.- 10.8 Solution of the Quartic Equation.- 10.9 Historical Commentary.- Exercises for Chapter 10.- Solutions to Some of the Exercises.- 11 Solvable Groups.- 11.1 First Definition.- 11.2 Derived or Commutator Subgroup.- 11.3 Second Definition of Solvability.- 11.4 Examples of Solvable Groups.- 11.5 Third Definition.- 11.6 The Group An Is Simple for n ? 5.- 11.6.1 Theorem.- 11.6.2 An Is Not Solvable for n ? 5, Direct Proof.- 11.7 Recent Results.- Exercises for Chapter 11.- Solutions to Some of the Exercises.- 12 Solvability of Equations by Radicals.- 12.1 Radical Extensions and Polynomials Solvable by Radicals.- 12.1.1 Radical Extensions.- 12.1.2 Polynomials Solvable by Radicals.- 12.1.3 First Construction.- 12.1.4 Second Construction.- 12.2 If a Polynomial Is Solvable by Radicals, Its Galois Group Is Solvable.- 12.3 Example of a Polynomial Not Solvable by Radicals.- 12.4 The Converse of the Fundamental Criterion.- 12.5 The General Equation of Degree n.- 12.5.1 Algebraically Independent Elements.- 12.5.2 Existence of Algebraically Independent Elements.- 12.5.3 The General Equation of Degree n.- 12.5.4 Galois Group of the General Equation of Degree n.- Exercises for Chapter 12.- Solutions to Some of the Exercises.- 13 The Life of Évariste Galois.- 14 Finite Fields.- 14.1 Algebraically Closed Fields.- 14.1.1 Definition.- 14.1.2 Algebraic Closures.- 14.1.3 Theorem (Steinitz, 1910).- 14.2 Examples of Finite Fields.- 14.3 The Characteristic of a Field.- 14.3.1 Definition.- 14.3.2 Properties.- 14.4 Properties of Finite Fields.- 14.4.1 Proposition.- 14.4.2 The Frobenius Homomorphism.- 14.5 Existence and Uniqueness of a Finite Field with pr Elements.- 14.5.1 Proposition.- 14.5.2 Corollary.- 14.6 Extensions of Finite Fields.- 14.7 Normality of a Finite Extension of Finite Fields.- 14.8 The Galois Group of a Finite Extension of a Finite Field.- 14.8.1 Proposition.- 14.8.2 The Galois Correspondence.- 14.8.3 Example.- Exercises for Chapter 14.- Solutions to Some of the Exercises.- 15 Separable Extensions.- 15.1 Separability.- 15.2 Example of an Inseparable Element.- 15.3 A Criterion for Separability.- 15.4 Perfect Fields.- 15.5 Perfect Fields and Separable Extensions.- 15.6 Galois Extensions.- 15.6.1 Definition.- 15.6.2 Proposition.- 15.6.3 The Galois Correspondence.- Toward Chapter 16.- 16 Recent Developments.- 16.1 The Inverse Problem of Galois Theory.- 16.1.1 The Problem.- 16.1.2 The Abelian Case.- 16.1.3 Example.- 16.2 Computation of Galois Groups over ? for Small-Degree Polynomials.- 16.2.1 Simplification of the Problem.- 16.2.2 The Irreducibility Problem.- 16.2.3 Embedding of G into Sn.- 16.2.4 Looking for G Among the Transitive Subgroups of Sn.- 16.2.5 Transitive Subgroups of S4.- 16.2.6 Study of ?(G) ? An.- 16.2.7 Study of ?(G) ? D4.- 16.2.8 Study of ?(G) ? ?/4?.- 16.2.9 An Algorithm for n = 4.
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J.-P. Escofier Galois Theory "Escofier's treatment, at a level suitable for advanced, senior undergraduates or first-year graduate students, centers on finite extensions of number fields, incorporating numerous examples and leaving aside finite fields and the entire concept of separability for the final chapters ... copious, well-chosen exercises ... are presented with their solutions ... The prose is ... spare and enthusiastic, and the proofs are both instructive and efficient ... Escofier has written an excellent text, offering a relatively elementary introduction to a beautiful subject in a book sufficiently broad to present a contemporary viewpoint and intuition but sufficiently restrained so as not to overwhelm the reader."--AMERICAN MATHEMATICAL SOCIETY
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