Galois Theory, Second Edition

Praise for the First Edition ". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!" —Monatshefte fur Mathematik Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galois theory of origami. In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including: The contributions of Lagrange, Galois, and Kronecker How to compute Galois groups Galois's results about irreducible polynomials of prime or prime-squared degree Abel's theorem about geometric constructions on the lemniscates Galois groups of quartic polynomials in all characteristics Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study. Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics. Contents 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