**History of Mathematics**

Volume: 39;
2012;
152 pp;
Softcover

MSC: Primary 01;

Print ISBN: 978-0-8218-8330-3

Product Code: HMATH/39

List Price: $49.00

AMS Member Price: $39.20

MAA member Price: $44.10

**Electronic ISBN: 978-0-8218-9033-2
Product Code: HMATH/39.E**

List Price: $49.00

AMS Member Price: $39.20

MAA member Price: $44.10

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#### Supplemental Materials

# Theory of Algebraic Functions of One Variable

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*Richard Dedekind; Heinrich Weber*

Translated and introduced by John Stillwell

A co-publication of the AMS and the London Mathematical Society

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 & Endorsements

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