Introductory algebraic number theoryPDF|Epub|txt|kindle电子书版本网盘下载

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1 Integral Domains1

1.1 Integral Domains1

1.2 Irreducibles and Primes5

1.3 Ideals8

1.4 Principal Ideal Domains10

1.5 Maximal Ideals and Prime Ideals16

1.6 Sums and Products of Ideals21

Exercises23

Suggested Reading25

Biographies25

2 Euclidean Domains27

2.1 Euclidean Domains27

2.2 Examples of Euclidean Domains30

2.3 Examples of Domains That are Not Euclidean37

2.4 Almost Euclidean Domains46

2.5 Representing Primes by Binary Quadratic Forms47

Exercises49

Suggested Reading51

Biographies53

3 Noetherian Domains54

3.1 Noetherian Domains54

3.2 Factorization Domains57

3.3 Unique Factorization Domains60

3.4 Modules64

3.5 Noetherian Modules67

Exercises71

Suggested Reading72

Biographies73

4 Elements Integral over a Domain74

4.1 Elements Integral over a Domain74

4.2 Integral Closure81

Exercises86

Suggested Reading87

Biographies87

5 Algebraic Extensions of a Field88

5.1 Minimal Polynomial of an Element Algebraic over a Field88

5.2 Conjugates of α over K90

5.3 Conjugates of an Algebraic Integer91

5.4 Algebraic Integers in a Quadratic Field94

5.5 Simple Extensions98

5.6 Multiple Extensions102

Exercises106

Suggested Reading108

Biographies108

6 Algebraic Number Fields109

6.1 Algebraic Number Fields109

6.2 Conjugate Fields of an Algebraic Number Field112

6.3 The Field Polynomial of an Element of an Algebraic Number Field116

6.4 The Discriminant of a Set of Elements in an Algebraic Number Field123

6.5 Basis of an Ideal129

6.6 Prime Ideals in Rings of Integers137

Exercises138

Suggested Reading140

Biographies140

7 Integral Bases141

7.1 Integral Basis of an Algebraic Number Field141

7.2 Minimal Integers160

7.3 Some Integral Bases in Cubic Fields170

7.4 Index and Minimal Index of an Algebraic Number Field178

7.5 Integral Basis of a Cyclotomic Field186

Exercises189

Suggested Reading191

Biographies193

8 Dedekind Domains194

8.1 Dedekind Domains194

8.2 Ideals in a Dedekind Domain195

8.3 Factorization into Prime Ideals200

8.4 Order of an Ideal with Respect to a Prime Ideal206

8.5 Generators of Ideals in a Dedekind Domain215

Exercises216

Suggested Reading217

9 Norms of Ideals218

9.1 Norm of an Integral Ideal218

9.2 Norm and Trace of an Element222

9.3 Norm of a Product of Ideals228

9.4 Norm of a Fractional Ideal231

Exercises233

Suggested Reading234

Biographies235

10 Factoring Primes in a Number Field236

10.1 Norm of a Prime Ideal236

10.2 Factoring Primes in a Quadratic Field241

10.3 Factoring Primes in a Monogenic Number Field249

10.4 Some Factorizations in Cubic Fields253

10.5 Factoring Primes in an Arbitrary Number Field257

10.6 Factoring Primes in a Cyclotomic Field260

Exercises261

Suggested Reading262

11 Units in Real Quadratic Fields264

11.1 The Units of Z ＋ Z？2264

11.2 The Equation x2 - my 2 = 1267

11.3 Units of Norm 1271

11.4 Units of Norm -1275

11.5 The Fundamental Unit278

11.6 Calculating the Fundamental Unit286

11.7 The Equation x2 - my 2 = N294

Exercises297

Suggested Reading298

Biographies298

12 The Ideal Class Group299

12.1 Ideal Class Group299

12.2 Minkowski’s Translate Theorem300

12.3 Minkowski’s Convex Body Theorem305

12.4 Minkowski’s Linear Forms Theorem306

12.5 Finiteness of the Ideal Class Group311

12.6 Algorithm to Determine the Ideal Class Group314

12.7 Applications to Binary Quadratic Forms331

Exercises341

Suggested Reading343

Biographies343

13 Dirichlet’s Unit Theorem344

13.1 Valuations of an Element of a Number Field344

13.2 Properties of Valuations346

13.3 Proof of Dirichlet’s Unit Theorem359

13.4 Fundamental System of Units361

13.5 Roots of Unity363

13.6 Fundamental Units in Cubic Fields369

13.7 Regulator378

Exercises382

Suggested Reading383

Biographies384

14 Applications to Diophantine Equations385

14.1 Insolvability of y2 = x3 ＋ k Using Congruence Considerations385

14.2 Solving y2 = x3 ＋ k Using Algebraic Numbers389

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

Exercises410

Suggested Reading411

Biographies411

List of Definitions413

Location of Theorems417

Location of Lemmas421

Bibliography423

Index425