On the Class Number of Abelian Number Fields -- Extended with Tables by Ken-ichi Yoshino and Mikihito Hirabayashi

With this translation, the classic monograph Über die Klassenzahl abelscher Zahlkörper by Helmut Hasse is now available in English for the first time. The book addresses three main topics: class number formulas for abelian number fields; expressions of the class number of real abelian number fields by the index of the subgroup generated by cyclotomic units; and the Hasse unit index of imaginary abelian number fields, the integrality of the relative class number formula, and the class number parity. Additionally, the book includes reprints of works by Ken-ichi Yoshino and Mikihito Hirabayashi, which extend the tables of Hasse unit indices and the relative class numbers to imaginary abelian number fields with conductor up to 100. The text provides systematic and practical methods for deriving class number formulas, determining the unit index and calculating the class number of abelian number fields. A wealth of illustrative examples, together with corrections and remarks on the original work, make this translation a valuable resource for today's students of and researchers in number theory. Foreword 5 Class Number Formulas 5 Computation of Class Numbers 6 Parity of Class Numbers 7 Structure of Class Groups 7 Class Number 1 Problems 8 Hilbert Class Fields 9 References 10 Translator's Preface 15 Preface to the Second Edition 18 Real Abelian Fields (Sects.2.1–2.12) 19 Relative Classes: The Hasse Index (Sects.3.2–3.8) 20 Relative Classes: Questions of Integrality (Sects.3.1 and 3.9–3.15) 21 Parity of the Class Number and Signature of the Unit Group (Sect.2.6) 21 Results Related to Invariant Classes (Sects.3.16–3.20) 21 Relative Classes: Numerical Results 22 Relative Classes and Stickelberger Elements 22 Small Class Numbers 23 Lower Bounds of Discriminants 23 Iwasawa's Theory 24 References 25 (a) Books and Expository Articles 25 (b) Articles 25 Preface to the First Edition 28 References 32 Contents 33 List of Theorems 35 Part I 37 Introduction 37 References 39 1 The Generalized Class Number Formulas 42 1.1 Abelian Number Fields as Class Fields 42 1.2 The Analytic Class Number Formula 44 1.3 Product Formulas for the Conductors and for the Gaussian Sums 46 1.4 Calculation of L-series 47 1.5 The Arithmetic Class Number Formula 50 1.6 Preliminary Remarks on the Arithmetic Structure of the Two Class Number Factors 52 References 55 2 The Arithmetic Structure of the Class Number Formula for Real Fields 57 2.1 Plan of Investigation 57 2.2 The First Way of Transformation 59 2.3 The Number Factor gK 62 2.4 Introduction to Cyclotomic Units 64 2.5 The First Arithmetic Representation of the Class Number 67 2.6 The Theorem of Weber and Its Generalization 71 2.7 Generalized Group Matrix 76 2.8 Linear Factor Decomposition of Generalized Group Determinant 79 2.9 The Number Factor cG 82 2.10 The Second Way of Transformation 86 2.11 The Second Arithmetic Representation of the Class Number 88 2.12 Real Cyclic Biquadratic Fields 91 References 96 3 The Arithmetic Structure of the Relative Class Number Formula for Imaginary Fields 98 3.1 Class-Field-Theoretic Proof of the Rational Integrality and the Arithmetic Meaning 98 3.2 The Unit Index Q 106 3.3 Criterion for Q = 1 or 2 by a Kummer-Generator 110 3.4 Criterion for Q = 1 or 2 by Ramification and Class Problem 114 3.5 Description of Ramification by the Characters 118 3.6 Criteria for Q = 1 or 2 by Characters and Class Problem 125 3.7 Types of Fields with Q = 1 and Types of Fields with Q = 2 127 3.8 Imaginary Bicyclic Biquadratic Fields 133 3.9 Preparation for Direct Proof of the Integrality 141 3.10 The Characters with Composite Conductor 144 3.11 Supplement of Gauss' Lemma 149 3.12 The Characters of 2-Power Order and with Composite Conductor 154 3.13 The Characters of Odd Prime-Power Conductor 158 3.14 The Characters of 2-Power Conductor 163 3.15 Direct Proof of the Integrality 166 3.16 Theorem of Weber and Its Supplement 173 Calculation of the Relative Class Numbers h2ρ* of P2ρ/P2ρ, 0 176 Calculation of the Relative Class Numbers h3ρ* of P3ρ/P3ρ, 0 181 Remarks on the General Case f(ψ) = pρ (p ≠2, ρ2) 190 3.17 Remarks on the Genus Factor 194 3.18 Divisibility by the Relative Class Number of a Subfield 195 3.19 Imaginary Abelian Number Fields with Odd Class Number 200 3.20 Imaginary Cyclic Fields with Odd Class Number 208 Appendix: Tables of Relative Class Numbers 221 Table of Contributions to Relative Class Number 221 Table of Relative Class Numbers 224 References 283 Part II 285 4 On the Relative Class Number of the Imaginary Abelian Number Field I 286 On the Relative Class Number of the Imaginary Abelian Number Field I 288 Introduction 288 1. Relative Class Number Formulae 289 2. Table of Unit Indices and Relative Class Numbers 293 Bibliography 335 5 On the Relative Class Number of the Imaginary Abelian Number Field II 337 On the Relative Class Number of the Imaginary Abelian Number Field III 339 Bibliography 387 6 Supplemental Readings 388 Bibliography on Hasse's Unit Indices of CM-Fields 388 Bibliography on Calculations of Relative Class Numbers of Imaginary Abelian Number Fields 389 Bibliography on Determinantal Expressions of Relative Class Numbers of Imaginary Abelian Number Fields 390 Index 394