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This book provides a selfcontained exposition of the theory of algebraic curves without requiring any of the prerequisites of modern algebraic geometry. The selfcontained treatment makes this important and mathematically central subject accessible to nonspecialists. At the same time, specialists in the field may be interested to discover several unusual topics. Among these are Tates theory of residues, higher derivatives and Weierstrass points in characteristic p, the StöhrVoloch proof of the Riemann hypothesis, and a treatment of inseparable residue field extensions. Although the exposition is based on the theory of function fields in one variable, the book is unusual in that it also covers projective curves, including singularities and a section on plane curves.
David Goldschmidt has served as the Director of the Center for Communications Research since 1991. Prior to that he was Professor of Mathematics at the University of California, Berkeley.  Reviews

From the reviews:
"This is a wellwritten book, which will quickly give the reader access to the theory of projective algebraic curves. The author manages to convey a very good amount of information on this subject, and there's also a lot of results on function fields. The treatment given to the theory of Weierstrass points, in which the ground field may have any characteristic, will certainly be remembered by the reader, even after he/she has studied the subject with the machinery offered by the scheme language. It is the opinion of this reviewer that this book is a fine contribution to a first study of algebraic functions and projective curves."  MATHEMATICAL REVIEWS
"This is a very nice algebraic introduction to the theory of algebraic curves (no geometry) with full, clear and simple proofs. It should be very useful for workers in coding theory." (Edoardo Ballico, Zentralblatt MATH, Vol. 1034, 2004)
"The author treats some topics not often found elsewhere like Tates theory of residues, inseparable residue field extensions, a proof of the Riemann hypothesis for finite fields etc. Since the book is rather selfcontained – even an appendix on field theory is provided – it can be recommended even for nonspecialists interested in this classical topic." (G. Kowol, Monatshefte für Mathematik, Vol. 143 (2), 2004)
"This book provides a selfcontained exposition of the theory of algebraic curves without requiring any of the prerequisites of modern algebraic geometry. The selfcontained treatment makes this important and mathematically central subject accessible to non specialists. At the same time, specialists in the field may be interested to discover several unusual topics." (L’ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (12), 2003)
"Goldschmidt … brings readers, in a minimal number of pages, from first principles to a major landmark of 20thcentury mathematics (which falls outside of Riemann surface theory!), namely, Weil’s Riemann hypothesis for curves over finite fields. An excellent stepping stone either to algebraic number theory or to abstract algebraic geometry." (D.V. Feldman, CHOICE, July 2003)
"The powerful interaction between algebra and geometry … led to an unprecedented development of many fields in mathematics, and in particular of the one presently called algebraic geometry. … This is a wellwritten book, which will quickly give the reader access to the theory of projective algebraic curves. The author manages to convey a very good amount of information on this subject … . this book is a fine contribution to a first study of algebraic functions and projective curves." (Cicero Fernandes de Carvalho, Mathematical Reviews, 2003 j)
 Table of contents (5 chapters)


Background
Pages 139

Function Fields
Pages 4067

Finite Extensions
Pages 68102

Projective Curves
Pages 103149

Zeta Functions
Pages 150163

Table of contents (5 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Algebraic Functions and Projective Curves
 Authors

 David Goldschmidt
 Series Title
 Graduate Texts in Mathematics
 Series Volume
 215
 Copyright
 2003
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9780387224459
 DOI
 10.1007/b97844
 Hardcover ISBN
 9780387954325
 Softcover ISBN
 9781441929952
 Series ISSN
 00725285
 Edition Number
 1
 Number of Pages
 XVI, 186
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