Introductory algebraic number theoryPDF电子书下载

1 Integral Domains 1

1.1 Integral Domains 1

1.2 Irreducibles and Primes 5

1.3 Ideals 8

1.4 Principal Ideal Domains 10

1.5 Maximal Ideals and Prime Ideals 16

1.6 Sums and Products of Ideals 21

Exercises 23

Suggested Reading 25

Biographies 25

2 Euclidean Domains 27

2.1 Euclidean Domains 27

2.2 Examples of Euclidean Domains 30

2.3 Examples of Domains That are Not Euclidean 37

2.4 Almost Euclidean Domains 46

2.5 Representing Primes by Binary Quadratic Forms 47

Exercises 49

Suggested Reading 51

Biographies 53

3 Noetherian Domains 54

3.1 Noetherian Domains 54

3.2 Factorization Domains 57

3.3 Unique Factorization Domains 60

3.4 Modules 64

3.5 Noetherian Modules 67

Exercises 71

Suggested Reading 72

Biographies 73

4 Elements Integral over a Domain 74

4.1 Elements Integral over a Domain 74

4.2 Integral Closure 81

Exercises 86

Suggested Reading 87

Biographies 87

5 Algebraic Extensions of a Field 88

5.1 Minimal Polynomial of an Element Algebraic over a Field 88

5.2 Conjugates of α over K 90

5.3 Conjugates of an Algebraic Integer 91

5.4 Algebraic Integers in a Quadratic Field 94

5.5 Simple Extensions 98

5.6 Multiple Extensions 102

Exercises 106

Suggested Reading 108

Biographies 108

6 Algebraic Number Fields 109

6.1 Algebraic Number Fields 109

6.2 Conjugate Fields of an Algebraic Number Field 112

6.3 The Field Polynomial of an Element of an Algebraic Number Field 116

6.4 The Discriminant of a Set of Elements in an Algebraic Number Field 123

6.5 Basis of an Ideal 129

6.6 Prime Ideals in Rings of Integers 137

Exercises 138

Suggested Reading 140

Biographies 140

7 Integral Bases 141

7.1 Integral Basis of an Algebraic Number Field 141

7.2 Minimal Integers 160

7.3 Some Integral Bases in Cubic Fields 170

7.4 Index and Minimal Index of an Algebraic Number Field 178

7.5 Integral Basis of a Cyclotomic Field 186

Exercises 189

Suggested Reading 191

Biographies 193

8 Dedekind Domains 194

8.1 Dedekind Domains 194

8.2 Ideals in a Dedekind Domain 195

8.3 Factorization into Prime Ideals 200

8.4 Order of an Ideal with Respect to a Prime Ideal 206

8.5 Generators of Ideals in a Dedekind Domain 215

Exercises 216

Suggested Reading 217

9 Norms of Ideals 218

9.1 Norm of an Integral Ideal 218

9.2 Norm and Trace of an Element 222

9.3 Norm of a Product of Ideals 228

9.4 Norm of a Fractional Ideal 231

Exercises 233

Suggested Reading 234

Biographies 235

10 Factoring Primes in a Number Field 236

10.1 Norm of a Prime Ideal 236

10.2 Factoring Primes in a Quadratic Field 241

10.3 Factoring Primes in a Monogenic Number Field 249

10.4 Some Factorizations in Cubic Fields 253

10.5 Factoring Primes in an Arbitrary Number Field 257

10.6 Factoring Primes in a Cyclotomic Field 260

Exercises 261

Suggested Reading 262

11 Units in Real Quadratic Fields 264

11.1 The Units of Z ＋ Z？2 264

11.2 The Equation x2 - my 2 = 1 267

11.3 Units of Norm 1 271

11.4 Units of Norm -1 275

11.5 The Fundamental Unit 278

11.6 Calculating the Fundamental Unit 286

11.7 The Equation x2 - my 2 = N 294

Exercises 297

Suggested Reading 298

Biographies 298

12 The Ideal Class Group 299

12.1 Ideal Class Group 299

12.2 Minkowski’s Translate Theorem 300

12.3 Minkowski’s Convex Body Theorem 305

12.4 Minkowski’s Linear Forms Theorem 306

12.5 Finiteness of the Ideal Class Group 311

12.6 Algorithm to Determine the Ideal Class Group 314

12.7 Applications to Binary Quadratic Forms 331

Exercises 341

Suggested Reading 343

Biographies 343

13 Dirichlet’s Unit Theorem 344

13.1 Valuations of an Element of a Number Field 344

13.2 Properties of Valuations 346

13.3 Proof of Dirichlet’s Unit Theorem 359

13.4 Fundamental System of Units 361

13.5 Roots of Unity 363

13.6 Fundamental Units in Cubic Fields 369

13.7 Regulator 378

Exercises 382

Suggested Reading 383

Biographies 384

14 Applications to Diophantine Equations 385

14.1 Insolvability of y2 = x3 ＋ k Using Congruence Considerations 385

14.2 Solving y2 = x3 ＋ k Using Algebraic Numbers 389

14.3 The Diophantine Equation y（y＋1）=x（x＋1）（x＋2） 401

Exercises 410

Suggested Reading 411

Biographies 411

List of Definitions 413

Location of Theorems 417

Location of Lemmas 421

Bibliography 423

Index 425