The Queen of Mathematics: A Historically Motivated Guide to Number Theory

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ISBN: 9781568810065 | 1568810067
Cover: Hardcover
Copyright: 11/15/1997

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The Queen of Mathematics takes the unique approach of examining number theory as it emerged in the 17th through 19th centuries. It leads to an understanding of today's research problems on the basis of their historical development. This book is a contribution to cultural history and brings a difficult subject within the reach of the serious reader.

PART 1: FROM FERMAT TO LEGENDRE

(85)

Chapter 1. The Founding Fathers

(11)

1. The Beginnings

(4)

2. Fermat's Mathematical Background

(3)

3. Pythagorean Triples

(3)

Appendix: Properties of the Integers

(1)

Chapter 2. Fermat

(12)

1. Fermat (1601-1665)

(1)

2. Infinite Descent

(1)

3. Fermat's Last Theorem

(3)

4. Pell's Equation

(2)

5. y(3) = x(2) + k

(1)

6. Sums of Squares

(2)

7. Perfect Numbers and Fermat's Little Theorem

(2)

8. Fermat's Error

(1)

Chapter 3. Euler

(19)

1. Euler

(3)

2. Partitions of a Number

(3)

3. The Beginning of Analytic Number Theory; Prime Numbers, Zeta Functions, Bernoulli Numbers

(8)

4. Arithmetic Functions

(1)

5. The Beginning of Algebraic Number Theory

(4)

Chapter 4. From Euler to Lagrange; The Theory of Continued Fractions

(33)

1. Introduction

(1)

2. The Basic Notions: Finite and Infinite Continued Fractions

(4)

The Continued Fraction Algorithm

(2)

3. The Early History

(1)

4. The Algebra of Finite Continued Fractions

(4)

5. The Arithmetic of Finite Continued Fractions

(3)

6. Infinite Continued Fractions

(3)

7. Diophantine Approximation and Geometry

(5)

8. Quadratic Irrationalities

(5)

9. Pell's Equation

(2)

10. Generalizations

(3)

Chapter 5. Lagrange

(5)

1. Lagrange and His Work

(2)

2. Quadratic Forms

(3)

Chapter 6. Legendre

(5)

1. Legendre

(1)

2. Rational Points on Conics

(1)

3. Distribution of Prime Numbers

(1)

4. Quadratic Residues and Quadratic Reciprocity

(3)

PART 2: GAUSS AND THE DISQUISITIONES ARITHMETICAE

(137)

Chapter 7. Gauss

(9)

1. Gauss and His Work

(8)

2. An Overview of the Disquisitiones Arithmeticae

(1)

Chapter 8. Theory of Congruence 1

(14)

1. Section I of the Disquisitiones

(4)

2. Residue Classes

(2)

3. Congruences and Algebraic Structures

101

(2)

4. Applications

103

(4)

5. Linear Congruences

107

(2)

Chapter 9. Theory of Congruences 2

109

(16)

1. Introduction

109

(1)

2. Reduced Residue Classes

109

(2)

3. The Structure of Z nZ

111

(3)

4. Polynomial Congruences

114

(5)

5. Polynomial Congruences and Polynomial Functions

119

(2)

6. Congruences in Several Variables; Chevally's Theorem

121

(2)

7. Solutions of Congruences and Solutions of Equations; The Hasse Principle

123

(2)

Chapter 10. Primitive Roots and Power Residues

125

(7)

1. Primitive Roots

125

(2)

2. Indices

127

(2)

3. k(th) Power Residues

129

(3)

Chapter 11. Congruences of the Second Degree

132

(23)

1. Introduction

132

(1)

2. Elementary Properties of Quadratic Residues

133

(2)

3. Gauss's Lemma

135

(1)

4. Computing (a/p)

136

(5)

5. Quadratic Reciprocity 1

141

(4)

6. Quadratic Reciprocity 2

145

(6)

7. Some History and Other Proofs

151

(4)

Chapter 12. Binary Quadratic Forms 1: Arithmetic Theory

155

(19)

1. Introduction

155

(1)

2. Equivalence of Forms

156

(1)

3. Matrix Notation and the Discriminant

157

(2)

4. Reduced Forms and the Number of Classes

159

(3)

5. Representation and Equivalence

162

(1)

6. Representations and Quadratic Residues

163

(3)

7. Proper Equivalence

166

(1)

8. Definite and Indefinite Forms

167

(2)

9. Positive Definite Forms

169

(2)

10. Primitive Forms and the Class Number

171

(3)

Chapter 13. Binary Quadratic Forms 2: Geometric Theory

174

(29)

1. Introduction

174

(1)

2. The Roots of a Form

175

(3)

3. Positive Definite Forms and the Upper Half Plane

178

(3)

4. Linear Fractional Transformations

181

(1)

5. The Fundamental Domain

182

(8)

6. Forms and the Upper Half Plane Revisited

190

(1)

7. Automorphs and the Number of Representations

191

(4)

8. Indefinite Forms, D greater than 0

195

(3)

Reduction

195

(1)

Automorphs and Representations

196

(1)

Geometric Methods

196

(2)

9. Composition of Forms

198

(3)

10. Genus

201

(1)

11. Sections V and VI of the Disquisitiones

202

(1)

Chapter 14. Cyclotomy

203

(20)

1. Introduction to Section VII of the Disquisitiones

203

(3)

2. Constructibility and the Theory of Equations

206

(3)

3. The 5-gon and Gaussian Periods

209

(3)

4. Back to Quadratic Reciprocity

212

(8)

5. Numbers of Solutions of Congruences; Equations over Finite Fields

220

(1)

6. Final Remarks on the Disquisitiones

221

(2)

PART 3: ALGEBRAIC NUMBER THEORY

223

(88)

Chapter 15. Algebraic Number Theory 1: The Gaussian Integers and Biquadratic Reciprocity

223

(18)

1. Gauss and Biquadratic Reciprocity

223

(4)

2. The Gaussian Integers

227

(3)

Back to the Two Square Problem

230

(2)

3. Congruence and the Law of Biquadratic Reciprocity

232

(4)

4. The Zeta Function and L Function of Z[i]

236

(5)

Chapter 16. Algebraic Number Theory 2: Algebraic Numbers and Quadratic Fields

241

(26)

1. The Development of Algebraic Number Theory

241

(9)

2. Algebraic Integers

250

(2)

3. Quadratic Fields

252

(2)

4. Quadratic Integers

254

(3)

5. Geometric Representation; Divisibility and Units

257

(3)

6. Factorization in Quadratic Fields

260

(1)

7. Euclidean Domains and Unique Factorization

261

(3)

8. Non-Unique Factorization and Ideals

264

(3)

Chapter 17. Algebraic Number Theory 3: Ideals in Quadratic Fields

267

(44)

1. Arithmetic of Ideals in I(d)

267

(3)

2. Lattices and Ideals

270

(5)

Appendix: Lattices

274

(1)

3. More Arithmetic of Ideals

275

(3)

4. Unique Factorization of Ideals

278

(3)

5. Applications of Unique Factorization

281

(1)

Divisibility and Diophantine Equations

281

(1)

Unique Factorization Domains

282

(1)

6. The Factorization of Rational Primes

282

(2)

7. Class Structure and the Class Number

284

(5)

8. Finiteness of the Class Number; Norm of an Ideal

289

(3)

9. Bases and Discriminants

292

(2)

10. The Correspondence between Forms and Fields

294

(5)

11. Applications of the Correspondence

299

(2)

Class Numbers

299

(1)

Units and Automorphs of Forms

300

(1)

The Class Group, Composition of Forms, and the Representation of Numbers by Forms

301

(1)

12. Factorization of Rational Primes Revisited

301

(6)

13. General Reciprocity Laws

307

(4)

Appendix: Dirichlet and 19th-Century Number Theory

309

(2)

PART 4: ARITHMETIC ON CURVES

311

(107)

Chapter 18. Arithmetic on Curves 1: Rational Points and Plane Algebraic Curves

311

(32)

1. Introduction

311

(2)

2. Lines

313

(1)

3. Conics

314

(5)

4. Cubics and the Geometric Form of Mordell's Theorem

319

(3)

5. The Need for Projective Geometry

322

(4)

6. The Real Projective Plane; Homogeneous Coordinates

326

(5)

The Real Projective Plane and its Models

326

(4)

Homogeneous Coordinates

330

(1)

7. Algebraic Curves in the Projective Plane

331

(6)

8. Geometry Over a Field; Higher Dimensions and Duality

337

(6)

General Fields

337

(2)

Higher Dimensions

339

(1)

Duality

339

(4)

Chapter 19. Arithmetic on Curves 2: Rational Points and Elliptic Curves

343

(40)

1. Introduction

343

(1)

2. Intersection of Curves; Bezout's Theorem

343

(5)

3. The Group Law and the Algebraic Form of Mordell's Theorem

348

(2)

4. Birational Equivalence; Weierstrass Normal Form

350

(5)

5. Singular Points and the Genus

355

(5)

6. Elliptic Curves and the Group Law

360

(8)

7. Elliptic Functions and Elliptic Curves

368

(4)

8. Complex Points of Finite Order

372

(1)

9. The Early History

373

(10)

Chapter 20. Arithmetic on Curves 3: The Twentieth Century

383

(35)

1. From Poincare to Weil

383

(5)

2. Points of Finite Order; The Lutz-Nagell Theorem

388

(2)

3. The Easy Part of the Theorem

390

(1)

4. The Hard Part of the Theorem

391

(8)

5. Mordell's Theorem; An Outline of the Proof

399

(3)

6. Some Preliminary Results

402

(2)

7. The Height Function

404

(5)

8. The Weak Mordell-Weil Theorem

409

(4)

9. Equations over Finite Fields; the Zeta and L Functions of a Curve

413

(3)

10. Complex Multiplication

416

(2)

PART 5: MISCELLANEOUS TOPICS

418

(79)

Chapter 21. Irrational and Transcendental Numbers, Diophantine Approximation

418

(22)

1. The Early History

418

(1)

2. From Euler to Dirichlet

419

(4)

3. Liouville to Hilbert; The Beginning of Transcendental Number Theory

423

(8)

4. Simultaneous Approximation; Kronecker's Theorem

431

(1)

5. Thue: Diophantine Approximation and Diophantine Equations

432

(4)

6. The 20th Century

436

(2)

7. Other Results and Problems

438

(1)

8. The Literature

439

(1)

Chapter 22. Geometry of Numbers

440

(30)

1. The Motivating Problem; Quadratic Forms

440

(3)

2. Minkowski's Fundamental Theorem

443

(5)

3. Minkowski's Theorem for Lattices

448

(4)

Lattices

448

(3)

Change of Basis

451

(1)

Minkowski's Theorem Reformulated

452

(1)

4. Back to Quadratic Forms

452

(2)

5. Sums of Two and Four Squares

454

(3)

6. Linear Forms

457

(2)

7. Sums and Products of Linear Forms; The Octahedron

459

(4)

8. Gauge Functions; The Equation of a Convex Body

463

(5)

9. Successive Minima

468

(1)

10. Other Directions

469

(1)

Chapter 23. p-adic Numbers and Valuations

470

(27)

1. History

470

(1)

2. The p-adic Numbers; An Informal Introduction

471

(7)

3. The Formal Development

478

(4)

4. Convergence

482

(5)

5. Congruences and p-adic Numbers

487

(2)

6. Hasse's Principle; The Hasse-Minkowski Theorem

489

(4)

7. Valuations and Algebraic Number Theory

493

(4)

Bibliography

497

(20)

Index

517

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The Queen of Mathematics: A Historically Motivated Guide to Number Theory

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