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.

Table of Contents

1 Foundations of the Theory of Algebraic Function Fields

2 Algebraic Geometry Codes

3 Extensions of Algebraic Function Fields

4 Differentials of Algebraic Function Fields

5 Algebraic Function Fields over Finite Constant Fields

6 Examples of Algebraic Function Fields

7 Asymptotic Bounds for the Number of Rational Places

8 More about Algebraic Geometry Codes

9 Subfield Subcodes and Trace Codes

Review

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)