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Theory of Algebraic Functions of One Variable
Richard Dedekind, Heinrich Weber Translated and introduced by John Stillwell
Price
1170.00
ISBN
9781470425906
Language
English
Pages
160
Format
Paperback
Dimensions
180 x 240 mm
Year of Publishing
2016
Territorial Rights
Restricted
Imprint
Universities Press

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, 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

Richard Dedekind, Heinrich Weber Translated and introduced by John Stillwell

Preface 
 Translator’s Introduction 
Overview 
From Calculus to 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 
Conclusion Theory of Algebraic Functions of One Variable Introduction 
Part I: Fields of algebraic functions 
Part II The points of the Riemann surface 
Bibliography 
Index

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