This book has two objectives. The first is to fill a void in the existing mathematical literature by providing a modern, self-contained and in-depth exposition of the theory of algebraic function fields. Topics include the Riemann-Roch theorem, algebraic extensions of function fields, ramifications theory and differentials. Particular emphasis is placed on function fields over a finite constant field, leading into zeta functions and the Hasse-Weil theorem. Numerous examples illustrate the general theory. Error-correcting codes are in widespread use for the reliable transmission of information. Perhaps the most fascinating of all the ties that link the theory of these codes to mathematics is the construction by V.D. Goppa, of powerful codes using techniques borrowed from algebraic geometry. Algebraic function fields provide the most elementary approach to Goppa's ideas, and the second objective of this book is to provide an introduction to Goppa's algebraic-geometric codes along these lines. The codes, their parameters and links with traditional codes such as classical Goppa, Peed-Solomon and BCH codes are treated at an early stage of the book. Subsequent chapters include a decoding algorithm for these codes as well as a discussion of their subfield subcodes and trace codes. Stichtenoth's book will be very useful to students and researchers in algebraic geometry and coding theory and to computer scientists and engineers interested in information transmission.
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Editorial Reviews
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From the reviews of the second edition: "In this book we have an exposition of the theory of function fields in one variable from the algebraic point of view … . The book is carefully written, the concepts are well motivated and plenty of examples help to understand the ideas and proofs and so it can be used as a textbook for an introductory course on the (classical) arithmetic of function fields with an application to coding theory." (Felipe Zaldivar, MAA Online, January, 2009)
--This text refers to an alternate
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From the Back Cover
The theory of algebraic function fields has its origins in number theory, complex analysis (compact Riemann surfaces), and algebraic geometry. Since about 1980, function fields have found surprising applications in other branches of mathematics such as coding theory, cryptography, sphere packings and others. The main objective of this book is to provide a purely algebraic, self-contained and in-depth exposition of the theory of function fields. This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded. Moreover, the present edition contains numerous exercises. Some of them are fairly easy and help the reader to understand the basic material. Other exercises are more advanced and cover additional material which could not be included in the text. This volume is mainly addressed to graduate students in mathematics and theoretical computer science, cryptography, coding theory and electrical engineering.
--This text refers to an alternate
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edition.
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3 of 3 people found the following review helpful:
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an introduction to Goppa codes,
By Mathieu Dutour (Jerusalem) - See all my reviews
This review is from: Algebraic Function Fields and Codes (Paperback)
Coding theory is fundamental to make digital transmission technology work efficiently and usually uses Reed-Solomon codes. The "natural" extension of those codes is to consider riemann surfaces over finite fields.The theory is developped from scratch and does not assume any knowledge of algebraic geometry. The author gave a proof of the Hasse-Weil bounds using the Zeta function. In parallel the theory of linear codes and Goppa codes is introduced from the beginning. While the author do not consider the geometry of riemann surfaces, having a knowledge of riemann surfaces over C can help a lot.
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Inside This Book
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First Sentence:
In this chapter we introduce the basic definitions and results of the theory of algebraic function fields: valuations, places, divisors, the genus of a function field, adeles, Weil differentials and the Riemann-Roch theorem. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
pairwise distinct places, full constant field, constant field extension, elliptic function field, holomorphy ring, subfield subcodes, pole divisor, finite constant field, algebraic function fields, finite separable extension, rational function field, trace codes, differential module, genus formula, complementary module, residue class field, integral basis, inertia field, divisor group, canonical divisor, principal divisor, valuation ring, irreducible monic polynomials, integral closure, inertia group Key Phrases - Capitalized Phrases (CAPs): (learn more)
Approximation Theorem, Strict Triangle Inequality, Hasse-Weil Theorem, Kummer's Theorem, Reed Solomon, Hurwitz Genus Formula, Hasse-Weil Bound, Serre Bound, Dedekind's Theorem, Proof of Step, Riemann's Theorem, Castelnuovo's Inequality, Dedekind's Different Theorem, Riemann's Inequality, Singleton Bound, Gilbert-Varshamov Bound, Proof of the Corollary, Proof of the Lemma, Hence Proposition, Proof of the Claim, Some Elementary Abelian, Using Proposition New!
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In this chapter we introduce the basic definitions and results of the theory of algebraic function fields: valuations, places, divisors, the genus of a function field, adeles, Weil differentials and the Riemann-Roch theorem. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
pairwise distinct places, full constant field, constant field extension, elliptic function field, holomorphy ring, subfield subcodes, pole divisor, finite constant field, algebraic function fields, finite separable extension, rational function field, trace codes, differential module, genus formula, complementary module, residue class field, integral basis, inertia field, divisor group, canonical divisor, principal divisor, valuation ring, irreducible monic polynomials, integral closure, inertia group Key Phrases - Capitalized Phrases (CAPs): (learn more)
Approximation Theorem, Strict Triangle Inequality, Hasse-Weil Theorem, Kummer's Theorem, Reed Solomon, Hurwitz Genus Formula, Hasse-Weil Bound, Serre Bound, Dedekind's Theorem, Proof of Step, Riemann's Theorem, Castelnuovo's Inequality, Dedekind's Different Theorem, Riemann's Inequality, Singleton Bound, Gilbert-Varshamov Bound, Proof of the Corollary, Proof of the Lemma, Hence Proposition, Proof of the Claim, Some Elementary Abelian, Using Proposition New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 68
books:
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77
books
cite this book:
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- Number theory, Volume 20 (Pure and Applied Mathematics) by Author Unknown in Back Matter
- Algebraic Numbers and Algebraic Functions (AMS Chelsea Publishing) by Emil Artin in Back Matter
- Dynamical Systems on Surfaces (Universitext) by Claude Godbillon in Back Matter
- Logic and Structure by Dirk van Dalen in Back Matter
- Geometric and Analytic Number Theory (Universitext) by Edmund Hlawka in Back Matter
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- Codes, Curves, and Signals: Common Threads in Communications (The Springer International Series in Engineering and Computer Science) by Alexander Vardy on 4 pages
- Finite Fields and Applications: Proceedings of the Third International Conference, Glasgow, July 1995 (London Mathematical Society Lecture Note Series) by S. Cohen on page 218, page 295, and page 378
- Rational Points on Curves over Finite Fields: Theory and Applications (London Mathematical Society Lecture Note Series) by Harald Niederreiter in Front Matter, and Back Matter
- Algorithmic Number Theory: Third International Symposium, ANTS-III, Portland, Orgeon, USA, June 21-25, 1998, Proceedings (Lecture Notes in Computer Science) by Joe P. Buhler on page 565, and page 606
- Number Theory and Its Applications (Lecture Notes in Pure and Applied Mathematics) by Serguei Stepanov on page 44, and page 258
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