Helmut Hasse - Böcker

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.

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From the reviews: "...a fine book ...treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer. This is not surprising, for it was he who inaugurated the local-global principle (universally called the Hasse principle). This doctrine asserts that one should first study a problem in algebraic number theory locally, that is, at the completion of a vaulation. Then ask for a miracle: that global validity is equivalent to local validity. Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924...The exposition is discursive...It is trite but true: Every number-theorist should have this book on his or her shelf." (Irving Kaplansky in Bulletin of the American Mathematical Society, 1981)

Inhalts}bersicht: Einleitung. - Die allgemeine Klassenzahlformel. - Die arithmetische Struktur der Klassenzahlformel f}r reelle K|rper. - Die arithmetische Struktur der Relativklassenzahlformel f}r imagin{re K|rper. - Anhang: Relativklassenzahltafeln. - Literaturverzeichnis.