History of valuation theory—Part I

The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kürschák who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory. In the present article which is planned as the first part of more to come, we report on the development of valuation theory until the ideas of Krull about general valuations of arbitrary rank took roots. That is, we cover the pre-Krull era. As our sources we use not only the published articles but also the information contained in letters and other material from that time, mostly but not exclusively from the legacy of Hasse at the University library at Göttingen. ∗This copy contains some minor corrections of the published version.