Galois theory / David A. Cox.

Format:

Book

Author/Creator:

Cox, David A.

Series:

Pure and applied mathematics (John Wiley & Sons : Unnumbered)

Pure and applied mathematics : a Wiley-Interscience series of texts, monographs, and tracts

Language:

English

Subjects (All):

Galois theory.

Physical Description:

xx, 559 pages : illustrations ; 25 cm.

Place of Publication:

Hoboken, N.J. : Wiley-Interscience, 2004.

Summary:

Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox's Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics. Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel's theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.

Contents:

Part I Polynomials

Chapter 1 Cubic Equations 3

1.1 Cardan's Formulas 3

1.2 Permutations of the Roots 10

A Permutations 10

B The Discriminant 11

C Symmetric Polynomials 13

1.3 Cubic Equations over the Real Numbers 15

A The Number of Real Roots 15

B Trigonometric Solution of the Cubic 18

Chapter 2 Symmetric Polynomials 25

2.1 Polynomials of Several Variables 25

A The Polynomial Ring in n Variables 25

B The Elementary Symmetric Polynomials 27

2.2 Symmetric Polynomials 30

A The Fundamental Theorem 30

B The Roots of a Polynomial 34

C Uniqueness 35

2.3 Computing with Symmetric Polynomials (Optional) 41

A Using Mathematica 42

B Using Maple 43

2.4 The Discriminant 46

Chapter 3 Roots of Polynomials 55

3.1 The Existence of Roots 55

3.2 The Fundamental Theorem of Algebra 62

Part II Fields

Chapter 4 Extension Fields 73

4.1 Elements of Extension Fields 73

A Minimal Polynomials 74

B Adjoining Elements 75

4.2 Irreducible Polynomials 81

A Using Maple and Mathematica 81

B Algorithms for Factoring 83

C The Schonemann-Eisenstein Criterion 84

D Prime Radicals 85

4.3 The Degree of an Extension 88

A Finite Extensions 89

B The Tower Theorem 91

4.4 Algebraic Extensions 94

Chapter 5 Normal and Separable Extensions 101

5.1 Splitting Fields 101

A Definitions and Examples 101

B Uniqueness 103

5.2 Normal Extensions 107

5.3 Separable Extensions 109

A Fields of Characteristic 0 112

B Fields of Characteristic p 113

C Computations 114

5.4 The Theorem of the Primitive Element 119

Chapter 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

A The pth Roots of 2 136

B The Universal Extension 137

C A Polynomial of Degree 5 138

6.5 Abelian Equations (Optional) 143

Chapter 7 The Galois Correspondence 147

7.1 Galois Extensions 147

A Splitting Fields of Separable Polynomials 147

B Finite Separable Extensions 150

C Galois Closures 151

7.2 Normal Subgroups and Normal Extensions 154

A Conjugate Fields 154

B Normal Subgroups 155

7.3 The Fundamental Theorem of Galois Theory 161

7.4 First Applications 167

A The Discriminant 167

B The Universal Extension 169

C The Inverse Galois Problem 170

7.5 Automorphisms and Geometry (Optional) 173

A Groups of Automorphisms 173

B Function Fields in One Variable 176

C Linear Fractional Transformations 178

D Stereographic Projection 180

Part III Applications

Chapter 8 Solvability by Radicals 191

8.1 Solvable Groups 191

8.2 Radical and Solvable Extensions 196

A Definitions and Examples 196

B Compositums and Galois Closures 198

C Properties of Radical and Solvable Extensions 198

8.3 Solvable Extensions and Solvable Groups 201

A Roots of Unity and Lagrange Resolvents 201

B Galois's Theorem 204

C Cardan's Formulas 207

8.4 Simple Groups 210

8.5 Solving Polynomials by Radicals 215

A Roots and Radicals 215

B The Universal Polynomial 217

C Abelian Equations 217

D The Fundamental Theorem of Algebra Revisited 218

8.6 The Casus Irreducibilis (Optional) 220

A Real Radicals 220

B Irreducible Polynomials with Real Radical Roots 222

C The Failure of Solvability in Characteristic p 224

Chapter 9 Cyclotomic Extensions 229

9.1 Cyclotomic Polynomials 229

A Some Number Theory 229

B Definition of Cyclotomic Polynomials 231

C The Galois Group of a Cyclotomic Extension 233

9.2 Gauss and Roots of Unity (Optional) 238

A The Galois Correspondence 238

B Periods 239

C Explicit Calculations 242

D Solvability by Radicals 246

Chapter 10 Geometric Constructions 255

10.1 Constructible Numbers 255

10.2 Regular Polygons and Roots of Unity 269

10.3 Origami (Optional) 273

A Origami Constructions 274

B Origami Numbers 276

C Marked Rulers and Intersections of Conics 279

Chapter 11 Finite Fields 289

11.1 The Structure of Finite Fields 289

A Existence and Uniqueness 289

B Galois Groups 292

11.2 Irreducible Polynomials over Finite Fields (Optional) 299

A Irreducible Polynomials of Fixed Degree 299

B Cyclotomic Polynomials Modulo p 301

C Berlekamp's Algorithm 303

Part IV Further Topics

Chapter 12 Lagrange, Galois, and Kronecker 313

12.1 Lagrange 313

A Resolvent Polynomials 314

B Similar Functions 318

C The Quartic 321

D Higher Degrees 324

E Lagrange Resolvents 326

12.2 Galois 332

A Beyond Lagrange 333

B Galois Resolvents 333

C Galois's Group 336

D Natural and Accessory Irrationalities 337

E Galois's Strategy 339

12.3 Kronecker 346

A Algebraic Quantities 346

B Module Systems 347

C Splitting Fields 349

Chapter 13 Computing Galois Groups 357

13.1 Quartic Polynomials 357

13.2 Quintic Polynomials 367

A Transitive Subgroups of S[subscript 5] 367

B Galois Groups of Quintics 370

D Solvable Quintics 376

13.3 Resolvents 384

A Jordan's Strategy 384

B Relative Resolvents 388

C Factoring Resolvents 389

13.4 Other Methods 394

A Kronecker's Analysis 394

B Dedekind's Theorem 398

Chapter 14 Solvable Permutation Groups 407

14.1 Polynomials of Prime Degree 407

14.2 Imprimitive Polynomials of Prime-Squared Degree 412

A Primitive and Imprimitive Groups 413

B Wreath Products 414

C The Solvable Case 417

14.3 Primitive Permutation Groups 422

A Doubly Transitive Permutation Groups 423

B Affine Linear and Semilinear Groups 424

C Minimal Normal Subgroups 425

D The Solvable Case 427

14.4 Primitive Polynomials of Prime-Squared Degree 438

A The First Two Subgroups 438

B The Third Subgroup 439

C The Solvable Case 444

Chapter 15 The Lemniscate 457

15.1 Division Points and Arc Length 458

A Division Points 458

B Arc Length of the Lemniscate 460

15.2 The Lemniscatic Function 465

A A Periodic Function 465

B Addition Laws 467

C Multiplication by Integers 470

15.3 The Complex Lemniscatic Function 476

A A Doubly Periodic Function 477

B Zeros and Poles 479

15.4 Complex Multiplication 483

A The Gaussian Integers 485

B Multiplication by Gaussian Integers 485

C Multiplication by Gaussian Primes 492

15.5 Abel's Theorem 499

A The Lemniscatic Galois Group 499

B Straightedge-and-Compass Constructions 500

Mathematical Notes 503

Historical Notes 504

Appendix A Abstract Algebra 509

A.1 Basic Algebra 509

A Groups 509

B Rings 513

C Fields 514

D Polynomials 515

A.2 Complex Numbers 518

A Addition, Multiplication, and Division 518

B Roots of Complex Numbers 519

A.3 Polynomials with Rational Coefficients 522

A.4 Group Actions 523

A The Sylow Theorems 526

B The Chinese Remainder Theorem 527

C The Multiplicative Group of a Field 527

D Unique Factorization Domains 528.

Notes:

Includes bibliographical references and index.

ISBN:

0471434191

OCLC:

54753018

Online:

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