Theory of Algebraic Functions of One Variable

The publication in 1882 of Dedekind and Weber’s paper on the algebraic approach to the theory of compact Riemann surfaces was one of those special moments in the history of mathematics. It opened an already important part of mathematics, the theory of functions, to the powerful algebraic methods that were being created at that time. But, perhaps more importantly, it also showed the deep relations between the arithmetic theory of algebraic number fields and the theory of functions fields in one variable. As it is explicit in the paper and in the introduction by the translator, Dedekind and Weber developed the subject following Dedekind’s approach to the arithmetic of algebraic fields in his Theory of Algebraic Integers, originally published in 1877 and also translated by J. Stillwell (Cambridge, 1996).

As is made clear by the Introduction and in the Summaries and Comments placed at the beginning of each of the 33 sections of the paper by the translator, the whole theory is developed only for the characteristic zero case, or to be more specific, for function fields over the complex numbers. We had to wait a few more decades until Artin, Hasse, Schmidt and others developed the corresponding theory of algebraic function fields in characteristic p. Along the way, and guided by the suggestive analogies between the arithmetic theory of number fields and the arithmetic of algebraic function fields over finite fields, Weil’s conjectures and program would eventually lead to a rewriting of the foundations of algebraic geometry in the second half of the 20^th century, a development whose consequences would be felt in almost all fields of mathematics.

The translation of this seminal paper, 130 years after its original publication, is a welcome opportunity to look at the roots of the subject, the algebraic part of geometry. With the annotations of the translator, and some fortunate choices (for example, the decision to use the familiar term vector space — see the footnote on page 59 — when the authors use the German term Schaar, whose English translation would be confusing to the modern reader), the paper is made easy to read and does not feel dated.

Some minor complaints: I would have preferred the use of the term effective or positive divisor instead of the misleading use of the word polygon (see the footnote in page 98), since these are terms that anyone would easily identify. Also, in the translator footnote of page 55, it is correct that the expression finitely generated module is well understood nowadays, and the translator contrasts this with the original term, finite module, used by authors. However in the algebraic geometry context, and given that Dedekind and Weber are working with algebras, the old term used by them is the one we use today: finite algebras are finitely generated as modules over the corresponding ring, finite-type algebras are finitely generated as algebras. I guess this is just an amusing turn of events in the use of terminology.

I enjoyed reading this translation and I am thankful to the AMS and LMS for their support and willingness to bring these foundational works to the modern reader. Stillwell has been enormously generous, sharing his mathematical and linguistic knowledge with us, and in addition to the works of Dedekind already mentioned, he has also translated Dirichlet’s Lectures on Number Theory (AMS, 1999), including some of the supplements added by Dedekind to this set of lectures.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.