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. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. Readership Undergraduate and graduate students and research mathematicians interested in algebra, algebraic geometry, and the history of mathematics. Reviews "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, the paper is made easy to read and does not feel dated. ... 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 [in] sharing his mathematical and linguistic knowledge with us." *-- Felipe Zaldivar, MAA Reviews* |