Galois Theory -- CONTENTS -- Preface to the First Edition -- Preface to the Second Edition -- Notation -- 1 Basic Notation -- 2 Chapter-by-Chapter Notation -- PART I POLYNOMIALS -- 1 Cubic Equations -- 1.1 Cardan's Formulas -- Historical Notes -- 1.2 Permutations of the Roots -- A Permutations -- B The Discriminant -- C Symmetric Polynomials -- Mathematical Notes -- Historical Notes -- 1.3 Cubic Equations over the Real Numbers -- A The Number of Real Roots -- B Trigonometric Solution of the Cubic -- Historical Notes -- References -- 2 Symmetric Polynomials -- 2.1 Polynomials of Several Variables -- A The Polynomial Ring in n Variables -- B The Elementary Symmetric Polynomials -- Mathematical Notes -- 2.2 Symmetric Polynomials -- A The Fundamental Theorem -- B The Roots of a Polynomial -- C Uniqueness -- Mathematical Notes -- Historical Notes -- 2.3 Computing with Symmetric Polynomials (Optional) -- A Using Mathematica -- B Using Maple -- 2.4 The Discriminant -- Mathematical Notes -- Historical Notes -- References -- 3 Roots of Polynomials -- 3.1 The Existence of Roots -- Mathematical Notes -- Historical Notes -- 3.2 The Fundamental Theorem of Algebra -- Mathematical Notes -- Historical Notes -- References -- PART II FIELDS -- 4 Extension Fields -- 4.1 Elements of Extension Fields -- A Minimal Polynomials -- B Adjoining Elements -- Mathematical Notes -- Historical Notes -- 4.2 Irreducible Polynomials -- A Using Maple and Mathematica -- B Algorithms for Factoring -- C The Schönemann-Eisenstein Criterion -- D Prime Radicals -- Historical Notes -- 4.3 The Degree of an Extension -- A Finite Extensions -- B The Tower Theorem -- Mathematical Notes -- Historical Notes -- 4.4 Algebraic Extensions -- Mathematical Notes -- References -- 5 Normal and Separable Extensions -- 5.1 Splitting Fields -- A Definition and Examples -- B Uniqueness.

5.2 Normal Extensions -- Historical Notes -- 5.3 Separable Extensions -- A Fields of Characteristic 0 -- B Fields of Characteristic p -- C Computations -- Mathematical Notes -- 5.4 Theorem of the Primitive Element -- Mathematical Notes -- Historical Notes -- References -- 6 The Galois Group -- 6.1 Definition of the Galois Group -- Historical Notes -- 6.2 Galois Groups of Splitting Fields -- 6.3 Permutations of the Roots -- Mathematical Notes -- Historical Notes -- 6.4 Examples of Galois Groups -- A The pth Roots of 2 -- B The Universal Extension -- C A Polynomial of Degree 5 -- Mathematical Notes -- Historical Notes -- 6.5 Abelian Equations (Optional) -- Historical Notes -- References -- 7 The Galois Correspondence -- 7.1 Galois Extensions -- A Splitting Fields of Separable Polynomials -- B Finite Separable Extensions -- C Galois Closures -- Historical Notes -- 7.2 Normal Subgroups and Normal Extensions -- A Conjugate Fields -- B Normal Subgroups -- Mathematical Notes -- Historical Notes -- 7.3 The Fundamental Theorem of Galois Theory -- 7.4 First Applications -- A The Discriminant -- B The Universal Extension -- C The Inverse Galois Problem -- Historical Notes -- 7.5 Automorphisms and Geometry (Optional) -- A Groups of Automorphisms -- B Function Fields in One Variable -- C Linear Fractional Transformations -- D Stereographic Projection -- Mathematical Notes -- References -- PART III APPLICATIONS -- 8 Solvability by Radicals -- 8.1 Solvable Groups -- Mathematical Notes -- 8.2 Radical and Solvable Extensions -- A Definitions and Examples -- B Compositums and Galois Closures -- C Properties of Radical and Solvable Extensions -- Historical Notes -- 8.3 Solvable Extensions and Solvable Groups -- A Roots of Unity and Lagrange Resolvents -- B Galois's Theorem -- C Cardan's Formulas -- Historical Notes -- 8.4 Simple Groups -- Mathematical Notes.

Historical Notes -- 8.5 Solving Polynomials by Radicals -- A Roots and Radicals -- B The Universal Polynomial -- C Abelian Equations -- D The Fundamental Theorem of Algebra Revisited -- Historical Notes -- 8.6 The Casus Irreducbilis (Optional) -- A Real Radicals -- B Irreducible Polynomials with Real Radical Roots -- C The Failure of Solvability in Characteristic p -- Historical Notes -- References -- 9 Cyclotomic Extensions -- 9.1 Cyclotomic Polynomials -- A Some Number Theory -- B Definition of Cyclotomic Polynomials -- C The Galois Group of a Cyclotomic Extension -- Historical Notes -- 9.2 Gauss and Roots of Unity (Optional) -- A The Galois Correspondence -- B Periods -- C Explicit Calculations -- D Solvability by Radicals -- Mathematical Notes -- Historical Notes -- References -- 10 Geometric Constructions -- 10.1 Constructible Numbers -- Mathematical Notes -- Historical Notes -- 10.2 Regular Polygons and Roots of Unity -- Historical Notes -- 10.3 Origami (Optional) -- A Origami Constructions -- B Origami Numbers -- C Marked Rulers and Intersections of Conics -- Mathematical Notes -- Historical Notes -- References -- 11 Finite Fields -- 11.1 The Structure of Finite Fields -- A Existence and Uniqueness -- B Galois Groups -- Mathematical Notes -- Historical Notes -- 11.2 Irreducible Polynomials over Finite Fields (Optional) -- A Irreducible Polynomials of Fixed Degree -- B Cyclotomic Polynomials Modulo p -- C Berlekamp's Algorithm -- Historical Notes -- References -- PART IV FURTHER TOPICS -- 12 Lagrange, Galois, and Kronecker -- 12.1 Lagrange -- A Resolvent Polynomials -- B Similar Functions -- C The Quartic -- D Higher Degrees -- E Lagrange Resolvents -- Historical Notes -- 12.2 Galois -- A Beyond Lagrange -- B Galois Resolvents -- C Galois's Group -- D Natural and Accessory Irrationalities -- E Galois's Strategy -- Historical Notes.

12.3 Kronecker -- A Algebraic Quantities -- B Module Systems -- C Splitting Fields -- Historical Notes -- References -- 13 Computing Galois Groups -- 13.1 Quartic Polynomials -- Mathematical Notes -- Historical Notes -- 13.2 Quintic Polynomials -- A Transitive Subgroups of S5 -- B Galois Groups of Quintics -- C Examples -- D Solvable Quintics -- Mathematical Notes -- Historical Notes -- 13.3 Resolvents -- A Jordan's Strategy -- B Relative Resolvents -- C Quartics in All Characteristics -- D Factoring Resolvents -- Mathematical Notes -- 13.4 Other Methods -- A Kronecker's Analysis -- B Dedekind's Theorem -- Mathematical Notes -- References -- 14 Solvable Permutation Groups -- 14.1 Polynomials of Prime Degree -- Mathematical Notes -- Historical Notes -- 14.2 Imprimitive Polynomials of Prime-Squared Degree -- A Primitive and Imprimitive Groups -- B Wreath Products -- C The Solvable Case -- Mathematical Notes -- Historical Notes -- 14.3 Primitive Permutation Groups -- A Doubly Transitive Permutation Groups -- B Affine Linear and Semilinear Groups -- C Minimal Normal Subgroups -- D The Solvable Case -- Mathematical Notes -- Historical Notes -- 14.4 Primitive Polynomials of Prime-Squared Degree -- A The First Two Subgroups -- B The Third Subgroup -- C The Solvable Case -- Mathematical Notes -- Historical Notes -- References -- 15 The Lemniscate -- 15.1 Division Points and Arc Length -- A Division Points of the Lemniscate -- B Arc Length of the Lemniscate -- Mathematical Notes -- Historical Notes -- 15.2 The Lemniscatic Function -- A A Periodic Function -- B Addition Laws -- C Multiplication by Integers -- Historical Notes -- 15.3 The Complex Lemniscatic Function -- A A Doubly Periodic Function -- B Zeros and Poles -- Mathematical Notes -- Historical Notes -- 15.4 Complex Multiplication -- A The Gaussian Integers -- B Multiplication by Gaussian Integers.

C Multiplication by Gaussian Primes -- Mathematical Notes -- Historical Notes -- 15.5 Abel's Theorem -- A The Lemniscatic Galois Group -- B Straightedge-and-Compass Constructions -- Mathematical Notes -- Historical Notes -- References -- A Abstract Algebra -- A.1 Basic Algebra -- A Groups -- B Rings -- C Fields -- D Polynomials -- A.2 Complex Numbers -- A Addition, Multiplication, and Division -- B Roots of Complex Numbers -- A.3 Polynomials with Rational Coefficients -- A.4 Group Actions -- A.5 More Algebra -- A The Sylow Theorems -- B The Chinese Remainder Theorem -- C The Multiplicative Group of a Field -- D Unique Factorization Domains -- B Hints to Selected Exercises -- C Student Projects -- References -- A Books and Monographs on Galois Theory -- B Books on Abstract Algebra -- C Collected Works -- Index.