Preface ix

**1 The Early History 1**

1.1 The Breakthrough 1

**2 Complex Numbers 9**

2.1 Rational Functions of Complex Numbers 9

2.2 Complex Roots 17

2.3 Solvability by Radicals I 23

2.4 Ruler and Compass Constructibility 26

2.5 Orders of Roots of Unity 36

2.6 The Existence of Complex Numbers* 38

**3 Solutions of Equations 45**

3.1 The Cubic Formula 45

3.2 Solvability by Radicals II 49

3.3 Other Types of Solutions* 50

**4 Modular Arithmetic 57**

4.1 Modular Addition, Subtraction, and Multiplication 57

4.2 The Euclidean Algorithm and Modular Inverses 62

4.3 Radicals in Modular Arithmetic* 69

4.4 The Fundamental Theorem of Arithmetic* 70

**5 The Binomial Theorem and Modular Powers 75**

5.1 The Binomial Theorem 75

5.2 Fermat's Theorem and Modular Exponents 85

5.3 The Multinomial Theorem* 90

5.4 The Euler **φ**-Function* 92

**6 Polynomials Over a Field 99**

6.1 Fields and Their Polynomials 99

6.2 The Factorization of Polynomials 107

6.3 The Euclidean Algorithm for Polynomials 113

6.4 Elementary Symmetric Polynomials* 119

6.5 Lagrange's Solution of the Quartic Equation* 125

**7 Galois Fields 131**

7.1 Galois's Construction of His Fields 131

7.2 The Galois Polynomial 139

7.3 The Primitive Element Theorem 144

7.4 On the Variety of Galois Fields* 147

**8 Permutations 155**

8.1 Permuting the Variables of a Function I 155

8.2 Permutations 158

8.3 Permuting the Variables of a Function II 166

8.4 The Parity of a Permutation 169

**9 Groups 183**

9.1 Permutation Groups 183

9.2 Abstract Groups 192

9.3 Isomorphisms of Groups and Orders of Elements 199

9.4 Subgroups and Their Orders 206

9.5 Cyclic Groups and Subgroups 215

9.6 Cayley's Theorem 218

**10 Quotient Groups and their Uses 225**

10.1 Quotient Groups 225

10.2 Group Homomorphisms 234

10.3 The Rigorous Construction of Fields 240

10.4 Galois Groups and Resolvability of Equations 253

**11 Topics in Elementary Group Theory 261**

11.1 The Direct Product of Groups 261

11.2 More Classifications 265

**12 Number Theory 273**

12.1 Pythagorean triples 273

12.2 Sums of two squares 278

12.3 Quadratic Reciprocity 285

12.4 The Gaussian Integers 293

12.5 Eulerian integers and others 304

12.6 What is the essence of primality? 310

**13 The Arithmetic of Ideals 317**

13.1 Preliminaries 317

13.2 Integers of a Quadratic Field 319

13.3 Ideals 322

13.4 Cancelation of Ideals 337

13.5 Norms of Ideals 341

13.6 Prime Ideals and Unique Factorization 343

13.7 Constructing Prime Ideals 347

**14 Abstract Rings 355**

14.1 Rings 355

14.2 Ideals 358

14.3 Domains 361

14.4 Quotients of Rings 367

A Excerpts: Al-Khwarizmi 377

B Excerpts: Cardano 383

C Excerpts: Abel 389

D Excerpts: Galois 395

E Excerpts: Cayley 401

F Mathematical Induction 405

G Logic, Predicates, Sets and Functions 413

G.1 Truth Tables 413

G.2 Modeling Implication 415

G.3 Predicates and their Negation 418

G.4 Two Applications 419

G.5 Sets 421

G.6 Functions 422

Biographies 427

Bibliography 431

Solutions to Selected Exercises 433

Index 440

Notation 444