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