计算代数数论研究生课程指南

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Contents

XVII

3.5.5. Discussion . . . . . . . . . . . . . . . . .

140

3.6. Additional Polynomial Algorithms. . . . . . . .

141

3.6.1. Modular Methods for Computing GCD's in

Z[X)

141

3.6.2. Factorization of Polynomials over a Number Field

142

3.6.3. A Root Finding Algorithm over C

145

3.7. Exercises for Chapter 3

.....

147

Chapter

Algorithms

for

Algebraic

Number

Theory

151

4.1. Algebraic Numbers and Number Fields

.......

151

4.1.1. Basic Definitions and Properties of Algebraic Numbers

151

4.1.2. Number Fields

.................

152

4.2. Representation and Operations on Algebraic Numbers .

156

4.2.1. Algebraic Numbers as Roots of their Minimal Polynomial.

156

4.2.2. The Standard Representation of an Algebraic Number . .

157

4.2.3. The Matrix (or Regular) Representation of an Algebraic Number

158

4.2.4. The Conjugate Vector Representation of an Algebraic Number

159

4.3. Trace, Norm and Characteristic Polynomial

.....

160

4.4.

Discriminants, Integral Bases and Polynomial Reduction

163

4.4.1. Discriminants and Integral Bases

163

4.4.2. The Polynomial Reduction Algorithm

166

4.5. The Subfield Problem and Applications

172

4.5.1. The Subfield Problem Using the

LLL

Algorithm

172

4.5.2. The Subfield Problem Using Linear Algebra over C

173

4.5.3. The Subfield Problem Using Algebraic Algorithms .

175

4.5.4. Applications of

the

Solutions to the Subfield Problem

177

4.6. Orders and Ideals .

179

4.6.1. Basic Definitions

179

4.6.2. Ideals of

184

4.7. Representation of Modules and Ideals

186

4.7.1. Modules and the Hermite Normal Form

186

4.7.2. Representation of Ideals . . . .

188

4.8. Decomposition of Prime Numbers I . . .

193

4.8.1. Definitions and Main Results . . . . .

194

4.8.2. A Simple Algorithm

for

the Decomposition of Primes

196

4.8.3. Computing Valuations . . . . .

198

4.8.4. Ideal Inversion and

the

Different .

202

4.9. Units and Ideal Classes

205

4.9.1. The Class Group

205

4.9.2. Units and the Regulator

206

4.9.3. Conclusion: the Main Computational Tasks of Algebraic

Number Theory . .

214

4.10. Exercises

for

Chapter 4 . . . . . . . . . . . . . . .

215

XVIII

Chapter

AlgorithIlls

for

Quadratic

Fields

5.1. Discriminant, Integral Basis and Decomposition of Primes

5.2. Ideals and Quadratic Forms

.........

5.3. Class Numbers of Imaginary Quadratic Fields

5.3.1. Computing Class Numbers Using Reduced Forms

5.3.2. Computing Class Numbers Using Modular Forms

5.3.3. Computing Class Numbers Using Analytic Formulas

5.4. Class Groups of Imaginary Quadratic Fields

5.4.1. Shanks's Baby Step Giant Step Method

5.4.2. Reduction and Composition of Quadratic Forms

5.4.3. Class Groups Using Shanks's Method

5.5. McCurley's Sub-exponential Algorithm

5.5.1. Outline of

the

Algorithm . . . . . .

5.5.2. Detailed Description of

the

Algorithm

5.5.3. Atkin's Variant . . . . . . . . . .

5.6. Class Groups of Real Quadratic Fields .

5.6.1. Computing Class Numbers Using Reduced Forms

5.6.2. Computing Class Numbers Using Analytic Formulas

5.6.3. A Heuristic Method of

Shanks.

. . . . . . . . .

Contents

218

220

226

229

232

235

238

245

247

250

255

257

261

263

5.7. Computation of

the

Fundamental Unit and of the Regulator 264

5.7.1. Description of

the

Algorithms . . . . . . . 264

5.7.2. Analysis of the Continued Fraction Algorithm

266

5.7.3. Computation of

the

Regulator.

. .

273

5.8.

The

Infrastructure Method of Shanks

274

5.8.1.

The

Distance Function . . . . . . 274

5.8.2. Description of

the

Algorithm . . .

278

5.8.3. Compact Representation of the Fundamental Unit 280

5.8.4. Other Application and Generalization of the Distance Function

282

5.9. Buchmann's Sub-exponential Algorithm . . . . . . . . .

283

5.9.1. Outline of

the

Algorithm . . . . . . . . . . . . . . .

284

5.9.2. Detailed Description of Buchmann's Sub-exponential Algorithm

286

5.10.

The

Cohen-Lenstra Heuristics . . . . . . . . . . . . 289

5.10.1. Results and Heuristics for Imaginary Quadratic

Fields.

290

5.10.2. Results and Heuristics for Real Quadratic Fields

292

5.11. Exercises for

Chapter

5 . . . . . . . . . . . . . . .

293

Chapter

6. AlgorithIllS for

Algebraic

NUIllber

Theory

297

6.1. Computing

the

Maximal Order . 297

6.1.1.

The

Pohst-Zassenhaus Theorem . 297

6.1.2.

The

Dedekind Criterion

. .

299

6.1.3. Outline of

the

Round 2 Algorithm 302

6.1.4. Detailed Description of the Round 2 Algorithm 305

6.2. Decomposition of Prime Numbers II . . . . . . 306

6.2.1. Newton Polygons . . . . . . . . . . . . . 307

6.2.2. Theoretical Description of the Buchmann-Lenstra Method 309

Contents

XIX

6.2.3. Multiplying and Dividing Ideals Modulo p . . . . . . .

311

6.2.4. Splitting of Separable Algebras over F p . • • . . . . . .

.•

312

6.2.5. Detailed Description of the Algorithm

for

Prime Decomposition 314

6.3. Computing Galois Groups

316

6.3.1. The Resolvent Method

316

6.3.2. Degree 3

319

6.3.3. Degree 4 319

6.3.4. Degree 5

322

6.3.5. Degree 6 323

6.3.6. Degree 7

325

6.3.7. A List of Test Polynomials 327

6.4. Examples of Families of Number Fields

328

6.4.1. Making Tables of Number Fields 328

6.4.2. Cyclic Cubic Fields

. . . . . . 330

6.4.3. Pure Cubic Fields . . . . . . . . . 337

6.4.4. Decomposition of Primes in Pure Cubic Fields

341

6.4.5. General Cubic

Fields.

. . . . . . . . . . .

345

6.5. Computing the Class Group, Regulator and Fundamental Units 346

6.5.1. Ideal Reduction . . . . . . . . . . . . . . . . . .

346

6.5.2. Computing the Relation Matrix . . . . . . . . . . .

348

6.5.3. Computing the Regulator and a System of Fundamental Units

351

6.5.4. The General Class Group and Unit

Algorithm.

352

6.5.5. The Principal Ideal Problem 354

6.6. Exercises for Chapter 6 . . . . . . . . . 356

Chapter

Introduction

Elliptic

Curves

360

7.1. Basic Definitions

7.1.1. Introduction

..........

7.1.2. Elliptic Integrals and Elliptic Functions

7.1.3. Elliptic Curves over a Field . . . . . .

7.1.4. Points on Elliptic Curves . . . . . . .

7.2. Complex Multiplication and Class Numbers

7.2.1. Maps Between Complex Elliptic Curves

7.2.2. Isogenies . . . . . . . . . . . . . . .

7.2.3. Complex Multiplication

.......

7.2.4. Complex Multiplication and Hilbert Class Fields

7.2.5. Modular Equations

.....

7.3. Rank and L-functions . . . . .

7.3.1. The Zeta Function of a Variety

7.3.2. L-functions of Elliptic Curves .

7.3.3. The Taniyama-Weil Conjecture

7.3.4. The Birch and Swinnerton-Dyer Conjecture

7.4. Algorithms for Elliptic Curves

.....

7.4.1. Algorithms for Elliptic Curves over C

7.4.2. Algorithm

for

Reducing a General Cubic

360

362

365

369

370

372

374

377

378

379

380

381

383

385

387

392

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