The Theory of Algebraic Number Fields

Contents

Translator's Preface
Hilbert's Preface
Introduction to the English Edition

Pt. I The Theory of General Number Fields

1 Algebraic Numbers and Number Fields 3
2 Ideals of Number Fields 9
3 Congruences with Respect to Ideals 17
4 The Discriminant of a Field and its Divisors 25
5 Extension Fields 33
6 Units of a Field 41
7 Ideal Classes of a Field 53
8 Reducible Forms of a Field 65
9 Orders in a Field 67

Pt. II Galois Number Fields

10 Prime Ideals of a Galois Number Field and its Subfields 79
11 The Differents and Discriminants of a Galois Number Field and its Subfields 89
12 Connexion Between the Arithmetic and Algebraic Properties of a Galois Number Field 93
13 Composition of Number Fields 97
14 The Prime Ideals of Degree 1 and the Class Concept 101
15 Cyclic Extension Fields of Prime Degree 105

Pt. III Quadratic Number Fields

16 Factorisation of Numbers in Quadratic Fields 115
17 Genera in Quadratic Fields and Their Character Sets 121
18 Existence of Genera in Quadratic Fields 133
19 Determination of the Number of Ideal Classes of a Quadratic Field 149
20 Orders and Modules of Quadratic Fields 155

Pt. IV Cyclotomic Fields

21 The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generate 161
22 The Roots of Unity for a Composite Exponent m and the Cyclotomic Field They Generate 167
23 Cyclotomic Fields as Abelian Fields 175
24 The Root Numbers of the Cyclotomic Field of the l-th Roots of Unity 187
25 The Reciprocity Law for l-th Power Residues Between a Rational Number and a Number in the Field of l-th Roots of Unity 199
26 Determination of the Number of Ideal Classes in the Cyclotomic Field of the m-th Roots of Unity 207
27 Applications of the Theory of Cyclotomic Fields to Quadratic Fields 217

Pt. V Kummer Number Fields

28 Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field 225
29 Norm Residues and Non-residues of a Kummer Field 233
30 Existence of Infinitely Many Prime Ideals with Prescribed Power Characters in a Kummer Field 253
31 Regular Cyclotomic Fields 257
32 Ambig Ideal Classes and Genera in Regular Kummer Fields 269
33 The l-th Power Reciprocity Law in Regular Cyclotomic Fields 289
34 The Number of Genera in a Regular Kummer Field 305
35 New Foundation of the Theory of Regular Kummer Fields 313
36 The Diophantine Equation [alpha][superscript m] + [beta][superscript m] + [gamma][superscript m] = 0 327

References 335
List of Theorems and Lemmas 345
Index 347