Dedekind and Weber: Theory of the algebraic functions of one variable

Dedekind and Weber, Theory of the algebraic functions of one variable

Editor’s note: My father, a lifelong scholar of mathematics, has recently completed a translation of a 19th century work by Dedekind and Weber from the original German to English. I offered him a place on this site to share his work as widely as possible.

What follows is his introduction to his translation, which is available here.

This seminal paper is much referenced in the literature, introducing as it does some basic concepts of modern algebra for an algebraically closed ground field of characteristic zero, and then applying them to Riemann surfaces etc. In conjunction with their other papers, (particularly those of Dedekind), this establishes some of the parallels with number theory.

The paper was written in 1880, in German. It was then something of a disappointment for me, as an amateur student, to find that after 130 years there still did not seem to be any English translation that I could use. I do not read German anything like sufficiently well to be able to follow a paper of this sort.

So, armed only with ancient schoolboy German (and the excellent on-line translation services from Google etc.) I set about producing some notes for my personal use. This was clearly going to take some time and, with the need to regularly revise as the mathematics suddenly made some usage clear, this was not going to be just a transcription job. So I thought that I might as well typeset as I went along, not least in order to get the multitude of equations and “fraktur” symbols looking reasonable.

It then occurred to me that there might be other students in a similar boat who want to get an appreciation of this major work but whose German is as limited as mine clearly is. So, when after several months I finally finished, I decided to make the result available at this site. If you come across this paper please feel free to make such use of it as you can (the pdf is set up for double-sided printing if you are not a screen junkie). It doubtless contains many mistakes and infelicities so beware and read it with your eyes open. If you feel kind enough to point out a selection of the errors of my ways then I would be most happy to receive an e-mail to that effect and to update the text accordingly.

Noel Robinson

(To e-mail: noel – at – gramlane – dot – fsnet – dot – co – dot – uk).

PS As I came to the end of this exercise, I became aware quite by chance that a far more erudite and professional rendering of this paper, including a commentary, will shortly be available. You wait 130 years for one to come along and then two come at once! Seriously though, these two translations serve different purposes. My limited effort is intended for students who might like to browse the paper to get an idea of what it was all about. (Who knew polygons could be such fun!). For definitive information you should as always turn to the professional.

PPS For similar reasons, I have another long term project on the back-burner which is to produce a working English version of SGA 4.5 by P. Deligne (from the French this time). If you are not interested in the beginnings of Etale Cohomology this is unlikely to excite you much. But if any supportive noises come my way, I will try and bring it to this site also.