Theory of Algebraic Functions of One Variable
About this Title
Richard Dedekind and Heinrich Weber. Translated by John Stillwell
Publication: History of Mathematics
Publication Year: 2012; Volume 39
ISBNs: 978-0-8218-8330-3 (print); 978-0-8218-9033-2 (online)
MathSciNet review: MR2962951
MSC: Primary 01A75; Secondary 01A55, 01A60, 14-03
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.
Undergraduate and graduate students and research mathematicians interested in algebra, algebraic geometry, and the history of mathematics.
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