Galois Theory

DAVID A. COX, PhD, is Professor in the Department of Mathematics at Amherst College. He has published extensively in his areas of research interest, which include algebraic geometry, number theory, and the history of mathematics. Dr. Cox is consulting editor for Wiley's Pure and Applied Mathematics book series and the author of Primes of the Form x2 + ny2 (Wiley).

“There is barely a better introduction to the subject, in all its theoretical and practical aspects, than the book under review.”  (Zentralblatt MATH, 1 December 2012)"the great merit of this book (one of many expositions of the subject) is that everything is taken at a slow pace, with many examples to illustrate every idea. You get the (probably true) impression that the author loves this material, has taught it to undergraduates at Amherst College many times, has learned by experience the ideas which students find difficult, and has then taken great trouble to dissect these ideas and to pick out exactly the right examples and exercises to make them part of the reader’s mental equipment." (The Mathematical Gazette 2016)

• Preface to the First Edition xviiPreface to the Second Edition xxiNotation xxiii1 Basic Notation xxiii2 Chapter-by-Chapter Notation xxvPART I POLYNOMIALS1 Cubic Equations 31.1 Cardan's Formulas 41.2 Permutations of the Roots 101.3 Cubic Equations over the Real Numbers 152 Symmetric Polynomials 252.1 Polynomials of Several Variables 252.2 Symmetric Polynomials 302.3 Computing with Symmetric Polynomials (Optional) 422.4 The Discriminant 463 Roots of Polynomials 553.1 The Existence of Roots 553.2 The Fundamental Theorem of Algebra 62PART II FIELDS4 Extension Fields 734.1 Elements of Extension Fields 734.2 Irreducible Polynomials 814.3 The Degree of an Extension 894.4 Algebraic Extensions 955 Normal and Separable Extensions 1015.1 Splitting Fields 1015.2 Normal Extensions 1075.3 Separable Extensions 1095.4 Theorem of the Primitive Element 1196 The Galois Group 1256.1 Definition of the Galois Group 1256.2 Galois Groups of Splitting Fields 1306.3 Permutations of the Roots 1326.4 Examples of Galois Groups 1366.5 Abelian Equations (Optional) 1437 The Galois Correspondence 1477.1 Galois Extensions 1477.2 Normal Subgroups and Normal Extensions 1547.3 The Fundamental Theorem of Galois Theory 1617.4 First Applications 1677.5 Automorphisms and Geometry (Optional) 173PART III APPLICATIONS8 Solvability by Radicals 1918.1 Solvable Groups 1918.2 Radical and Solvable Extensions 1968.3 Solvable Extensions and Solvable Groups 2018.4 Simple Groups 2108.5 Solving Polynomials by Radicals 2158.6 The Casus Irreducbilis (Optional) 2209 Cyclotomic Extensions 2299.1 Cyclotomic Polynomials 2299.2 Gauss and Roots of Unity (Optional) 23810 Geometric Constructions 25510.1 Constructible Numbers 25510.2 Regular Polygons and Roots of Unity 27010.3 Origami (Optional) 27411 Finite Fields 29111.1 The Structure of Finite Fields 29111.2 Irreducible Polynomials over Finite Fields (Optional) 301PART IV FURTHER TOPICS12 Lagrange, Galois, and Kronecker 31512.1 Lagrange 31512.2 Galois 33412.3 Kronecker 34713 Computing Galois Groups 35713.1 Quartic Polynomials 35713.2 Quintic Polynomials 36813.3 Resolvents 38613.4 Other Methods 40014 Solvable Permutation Groups 41314.1 Polynomials of Prime Degree 41314.2 Imprimitive Polynomials of Prime-Squared Degree 41914.3 Primitive Permutation Groups 42914.4 Primitive Polynomials of Prime-Squared Degree 44415 The Lemniscate 46315.1 Division Points and Arc Length 46415.2 The Lemniscatic Function 47015.3 The Complex Lemniscatic Function 48215.4 Complex Multiplication 48915.5 Abel's Theorem 504A Abstract Algebra 515A.1 Basic Algebra 515A.2 Complex Numbers 524A.3 Polynomials with Rational Coefficients 528A.4 Group Actions 530A.5 More Algebra 532Index 557