Galois''s results about irreducible polynomials of prime orprime-squared degre
Trade Review
“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)
Table of 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
