This book is the first English translation of the classic long paper
Theorie der algebraischen Functionen einer Veränderlichen (Theory of
algebraic functions of one variable), published by Dedekind and Weber
in 1882. The translation has been enriched by a Translator's
Introduction that includes historical background, and also by extensive
commentary embedded in the translation itself.
The translation, introduction, and commentary provide the first
easy access to this important paper for a wide mathematical audience:
students, historians of mathematics, and professional
mathematicians.
Why is the Dedekind-Weber paper important? In the 1850s, Riemann
initiated a revolution in algebraic geometry by interpreting algebraic
curves as surfaces covering the sphere. He obtained deep and striking
results in pure algebra by intuitive arguments about surfaces and
their topology. However, Riemann's arguments were not rigorous, and
they remained in limbo until 1882, when Dedekind and Weber put them on
a sound foundation.
The key to this breakthrough was to develop the theory of algebraic
functions in analogy with Dedekind's theory of algebraic numbers,
where the concept of ideal plays a central role. By introducing such
concepts into the theory of algebraic curves, Dedekind and Weber paved
the way for modern algebraic geometry.
Readership
Undergraduate and graduate students and research
mathematicians interested in algebra, algebraic geometry, and the
history of mathematics.