Galois theory

Preface to the First Edition xvii     Preface to the Second Edition xxi     Notation xxiii     1 Basic Notation xxiii     2 Chapter-by-Chapter Notation xxv     PART I POLYNOMIALS     1 Cubic Equations 3     1.1 Cardan's Formulas 4     1.2 Permutations of the Roots 10     1.3 Cubic Equations over the Real Numbers 15     2 Symmetric Polynomials 25     2.1 Polynomials of Several Variables 25     2.2 Symmetric Polynomials 30     2.3 Computing with Symmetric Polynomials (Optional) 42     2.4 The Discriminant 46     3 Roots of Polynomials 55     3.1 The Existence of Roots 55     3.2 The Fundamental Theorem of Algebra 62     PART II FIELDS     4 Extension Fields 73     4.1 Elements of Extension Fields 73     4.2 Irreducible Polynomials 81     4.3 The Degree of an Extension 89     4.4 Algebraic Extensions 95     5 Normal and Separable Extensions 101     5.1 Splitting Fields 101     5.2 Normal Extensions 107     5.3 Separable Extensions 109     5.4 Theorem of the Primitive Element 119     6 The Galois Group 125     6.1 Definition of the Galois Group 125     6.2 Galois Groups of Splitting Fields 130     6.3 Permutations of the Roots 132     6.4 Examples of Galois Groups 136     6.5 Abelian Equations (Optional) 143     7 The Galois Correspondence 147     7.1 Galois Extensions 147     7.2 Normal Subgroups and Normal Extensions 154     7.3 The Fundamental Theorem of Galois Theory 161     7.4 First Applications 167     7.5 Automorphisms and Geometry (Optional) 173     PART III APPLICATIONS     8 Solvability by Radicals 191     8.1 Solvable Groups 191     8.2 Radical and Solvable Extensions 196     8.3 Solvable Extensions and Solvable Groups 201     8.4 Simple Groups 210     8.5 Solving Polynomials by Radicals 215     8.6 The Casus Irreducbilis (Optional) 220     9 Cyclotomic Extensions 229     9.1 Cyclotomic Polynomials 229     9.2 Gauss and Roots of Unity (Optional) 238     10 Geometric Constructions 255     10.1 Constructible Numbers 255     10.2 Regular Polygons and Roots of Unity 270     10.3 Origami (Optional) 274     11 Finite Fields 291     11.1 The Structure of Finite Fields 291     11.2 Irreducible Polynomials over Finite Fields (Optional) 301     PART IV FURTHER TOPICS     12 Lagrange, Galois, and Kronecker 315     12.1 Lagrange 315     12.2 Galois 334     12.3 Kronecker 347     13 Computing Galois Groups 357     13.1 Quartic Polynomials 357     13.2 Quintic Polynomials 368     13.3 Resolvents 386     13.4 Other Methods 400     14 Solvable Permutation Groups 413     14.1 Polynomials of Prime Degree 413     14.2 Imprimitive Polynomials of Prime-Squared Degree 419     14.3 Primitive Permutation Groups 429     14.4 Primitive Polynomials of Prime-Squared Degree 444     15 The Lemniscate 463     15.1 Division Points and Arc Length 464     15.2 The Lemniscatic Function 470     15.3 The Complex Lemniscatic Function 482     15.4 Complex Multiplication 489     15.5 Abel's Theorem 504     A Abstract Algebra 515     A.1 Basic Algebra 515     A.2 Complex Numbers 524     A.3 Polynomials with Rational Coefficients 528     A.4 Group Actions 530     A.5 More Algebra 532     Index 557