Helmut Hasse Livres

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)

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.

This book contains the full text of the mathematical notebooks of Helmut Hasse (1898-1979), who was one of the leading mathematicians of the 20th century. The originals have been preserved in the Handschriftenabteilung at the Göttingen University Library. There are a total of seven volumes which contain 98 entries all together; these span the time period from July 1923 to February 1935. Each of the entries is preceded by a short comment. An extensive bibliography is provided. Compared to the other documents and papers from Hasse Nachlass at Göttingen, these papers are quite different. They detail Hasse‘s mathematical background, interests and his way of approaching problems. Many of the entries are inspired by Hasse‘s discussions with other mathematicians (for example, Emil Artin is often mentioned). One of the notable highlights in the entries is the fi rst documentation of Artin‘s conjecture on primitive roots (1927), which Hasse noted down after a discussion with Artin. Another example is the fi rst proof of the Riemann hypothesis for an extended class of function fi elds of high genus, namely those of the generalised Fermat fi elds with a fi nite fi eld of constants (1932), after corresponding with Harold Davenport.

Das Inhaltsverzeichnis bietet eine strukturierte Übersicht über die Themen und Kapitel eines Buches. Es ermöglicht den Lesern, schnell relevante Abschnitte zu finden und sich einen Überblick über den Aufbau und die Inhalte zu verschaffen. Durch die klare Gliederung wird die Navigation innerhalb des Werkes erleichtert, was besonders hilfreich ist, um spezifische Informationen gezielt zu suchen.

Die Entwicklung der algebraischen Zahlentheorie wird detailliert beschrieben, beginnend mit den Grundlagen, die Hilbert in seinem Bericht darlegte. Dieser umfasst allgemeine Aspekte sowie spezielle Typen algebraischer Zahlkörper, darunter quadratische, Kreiskörper und Kummersche Zahlkörper. Die zweite Phase, die sich auf relativ-abelsche Zahlkörper konzentriert, wird durch Hilberts Konzept der Klassenkörpertheorie geprägt. Trotz der umfassenden theoretischen Fortschritte ist der Wunsch nach konkreten Beispielen und einer expliziten Behandlung der Themen in den Hintergrund gerückt.

This book reproduces the complete extant correspondence between Emmy Noether and Helmut Hasse. There are 82 such letters, of which 79 are from Noether to Hasse, dating from 1925 until Noether's sudden death in 1935. The correspondence reflects a crucial period in the development of 20th century algebra and number theory, in particular class field theory. Details of proofs appear alongside with conjectures and speculations. Also discussed are questions of textbook presentation, e. g., of Galois theory. Aside from mathematical details, the spontaneity of Noether's style allows many glimpses at the image that Emmy Noether and Helmut Hasse had of the topics they were working in. The Hasse - Noether correspondence is a rich source for those who are interested in the rise and the development of mathematical notions and ideas. Each letter is accompanied by a detailed commentary supplied by the editors. For the convenience of the reader, numerous cross-references, extended indexes, and short biographies of all persons mentioned in the correspondence have been added.