×

Developments in the theory of algebras over number fields: a new foundation for the Hasse norm residue symbol and new approaches to both the Artin reciprocity law and class field theory. (English) Zbl 1151.01009

Gray, Jeremy J. (ed.) et al., Episodes in the history of modern algebra (1800–1950). Based on a workshop held at MSRI, Berkeley, CA, USA, April 2003. Providence, RI: American Mathematical Society (AMS); London: London Mathematical Society (ISBN 978-0-8218-4343-7/hbk). History of Mathematics 32, 117-151 (2007).
The paper first reviews some of the main events pertinent for the sequel that led from Hamilton’s discovery of the quaternions in 1843 to the books by L. E. Dickson, Algebras and their arithmetics (1923; JFM 49.0079.01) and Algebren und ihre Zahlentheorie (1927; JFM 53.0112.01). Then a general indication is given of the influence these had, mainly on the work of Hasse, but also on that of Artin, Emmy Noether, A. Adrian Albert, and Richard Brauer. It is also showed how this led to Hasse’s new approach to the norm residue symbol, and to class field theory. The parts of the paper are: The Beginnings of Structure Theory, Wedderburn’s General Structure Theorems, Hurwitz and the Arithmetic of Quaternions, The Structure of Skew Fields: Connections with Algebraic Number Theory, The Theory of Semisimple Algebras, The Local Theory and the Theorem of Brauer-Hasse-Noether, The New Norm Residue Symbol and New Approaches to Both the Reciprocity Law and Class Field Theory.
For the entire collection see [Zbl 1117.01001].

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
11-03 History of number theory
PDFBibTeX XMLCite