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.

First English translation of the original German edition Provides systematic tables on the relative class number of imaginary abelian number fields Contains rigorous results on the class numbers of abelian number fields Offers numerous, deeper, algebraic and number-theoretic details Suitable for students of algebraic number theory

Autorentext
Biography of Helmut Hasse (1898-1979) Born on August 25, 1898 in Kassel, Germany, Helmut Hasse studied at the University of Göttingen after WWI. Of his teachers there including Landau, Hilbert and Ehmy Noether, Hecke influenced him most. In 1820, Hasse went to Marburg, and under the direction of Kurt Hensel, discovered what is now known as the Hasse principle, or "local-global" principle, in algebraic number theory. He held further positions at the universities in Kiel and Hall prior to 1933. With the troubles of 1933, Hermann Weyl, who had succeeded Hilbert in the foremost chair or mathematics in Germany, resigned and Helmut Hasse was appointed in this place. The following year, Hasse became director of the Mathematical Institute at Göttingen. From 1939 to 1945, Hasse worked in Berlin for the navy on problems in ballistics. He returned to Göttingen but was soon dismissed by the British occupation forces. In 1946 he took a research position at the Berlin Academy. Thereafter, he held positions at the Humboldt University in East Berlin, and, from 1950 until retirement in 1966, at the University of Hamburg. At Halle, Hasse obtained fundamental results on the structure of central simple algebras over local fields. In Marburg, he did joint work with Brauer and Emmy Noether on simple algebras, also on elliptic curves and topological fields. In particular, he proved the analogon of the Riemann Hypothesis for zeta functions of elliptic curves. Both of Hasse s famous books Über die Klassenzahl abelscher Zahlkörper und Zahlentheorie appeared during his years in Berlin.

Inhalt
Part I .- Introduction.- The Generalized Class Number Formulas.- The Arithmetic Structure of the Class Number Formula for Real Fields.- The Arithmetic Structure of the Relative Class Number Formula for Imaginary Fields.- Appendix: Tables of Relative Class Numbers - Part II .- On the Relative Class Number of the Imaginary Abelian Number Field I.- On the Relative Class Number of the Imaginary Abelian Number Field II.- Supplemental Readings.