Theory of algebraic functions of one variable. Transl. from the German and introduced by John Stillwell (Q2897223)

Exactly 130 years ago, in 1882, Richard Dedekind (1831--1916) and Heinrich Weber (1842--1913) published their fundamental treatise ``Theorie der algebraischen Functionen einer Veränderlichen'' in [Kronecker J. XCII, 181--291 (1882; JFM 14.0352.01)]. This seminal paper, which was extensively reviewed by Emil Toeplitz at that time, turned out to be one of the most influential works in the history of algebraic geometry. In fact, Dedekind and Weber virtually laid the foundations of modern, contemporary algebraic geometry, mainly by introducing the new concepts and methods from algebraic number theory to the theory of algebraic curves. More precisely, they used the frame work of rings and ideals to give rigorous algebraic proofs of results on curves, which were previously derived, in a merely intuitive fashion, by topological and complex-analytical reasoning for Riemann surfaces. In this way, the authors' 1882 paper established a rigorous, purely algebraic approach to the birational geometry of algebraic curves, thereby exhibiting the deep analogy between number fields and algebraic function fields. The development of algebraic geometry in the course of the 20th century provides clear evidence of the pioneering character of this paper, and thus it continues to be cited in the contemporary literature. However, for about 130 years, this paper only existed in the mathematical language of the 1880s, which fewer mathematicians today are able (or willing) to deal with, and in the old-fashioned German language at that time, which is hard to understand even for Germans in these days.NEWLINENEWLINE The booklet under review is the overdue English translation of the Dedekind-Weber paper from 1882. John Stillwell has not only provided this excellent translation of the original, but also equipped it with useful, clarifying comments to assist the modern reader. His main commentary appears in the form of a ``Translator's Introduction'', which occupies the first thirty-seven pages of the book. In this part, J. Stillwell explains the historical background to the Dedekind-Weber paper, including sketches of the following topics:NEWLINENEWLINE Abel's theory of algebraic curves; Riemann's theory of algebraic curves; the Riemann-Hurwitz formula; functions on Riemann surfaces; later development of analysis on Riemann surfaces; origins of algebraic number theory; Dedekind's theory of algebraic integers; number fields and function fields; algebraic functions and Riemann surfaces; from points to valuations; reading the Dedekind-Weber paper; and conclusion (from this introduction).NEWLINENEWLINE It follows then the translation of the German original of the Dedekind Weber paper from 1882, which consists of two major parts and thirty-three sections altogether. J. Stillwell has enriched this part by many additional comments and numerous footnotes inserted in the translation itself. More precisely, the comments in the beginning of each section are to guide the reader through the (somewhat terse and rather difficult) original text, while the footnotes mainly concern specific details and questions of terminology.NEWLINENEWLINE At the end of the book, there is a carefully compiled bibliography, which mainly refers to the ``Translator's Introduction'' and to the other comments. A just as carefully arranged index helps the reader work with the present translation of the Dedekind-Weber paper in a convenient way, and the whole printing looks very pleasing as well.NEWLINENEWLINE All together, the translator (J. Stillwell) has done a great service to the mathematical community as a whole. Finally, the important paper by R. Dedekind and H. Weber from 1882 has been made both available and mathematically accessible to further generations of algebraic geometers, historians of mathematics, and historically interested students.