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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. The Generalized Class Number Formulas

Abstract
I begin with a brief summary of the essential class-field-theoretic facts on abelian number fields for all the rest of this book, most of which, as we will see, can be found in detail in my Klassenzahlbericht, or arise as special cases of abelian number fields by general theorems on the class field. Here I restrict myself to main facts.
Helmut Hasse

### Chapter 2. The Arithmetic Structure of the Class Number Formula for Real Fields

Abstract
In this second chapter we always assume that an abelian number field K we consider is real and hence coincides with its maximal real subfield K 0.
Helmut Hasse

### Chapter 3. The Arithmetic Structure of the Relative Class Number Formula for Imaginary Fields

Abstract
In this third chapter we always assume that an abelian number field K we consider is imaginary and hence quadratic over its maximal real subfield K 0.
Helmut Hasse

Without Abstract
Helmut Hasse

Without Abstract
Helmut Hasse