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Algebraic Function Fields and Codes

Algebraic Function Fields and Codes
Author: Henning Stichtenoth
Publisher: Springer Science & Business Media
Total Pages: 360
Release: 2009-02-11
Genre: Mathematics
ISBN: 3540768785

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This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.


Topics in the Theory of Algebraic Function Fields

Topics in the Theory of Algebraic Function Fields
Author: Gabriel Daniel Villa Salvador
Publisher: Springer Science & Business Media
Total Pages: 658
Release: 2007-10-10
Genre: Mathematics
ISBN: 0817645152

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The fields of algebraic functions of one variable appear in several areas of mathematics: complex analysis, algebraic geometry, and number theory. This text adopts the latter perspective by applying an arithmetic-algebraic viewpoint to the study of function fields as part of the algebraic theory of numbers. The examination explains both the similarities and fundamental differences between function fields and number fields, including many exercises and examples to enhance understanding and motivate further study. The only prerequisites are a basic knowledge of field theory, complex analysis, and some commutative algebra.


Algebraic Functions and Projective Curves

Algebraic Functions and Projective Curves
Author: David Goldschmidt
Publisher: Springer Science & Business Media
Total Pages: 196
Release: 2003
Genre: Mathematics
ISBN: 0387954325

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This book grew out of a set of notes for a series of lectures I orginally gave at the Center for Communications Research and then at Princeton University. The motivation was to try to understand the basic facts about algebraic curves without the modern prerequisite machinery of algebraic geometry. Of course, one might well ask if this is a good thing to do. There is no clear answer to this question. In short, we are trading off easier access to the facts against a loss of generality and an impaired understanding of some fundamental ideas. Whether or not this is a useful tradeoff is something you will have to decide for yourself. One of my objectives was to make the exposition as self-contained as possible. Given the choice between a reference and a proof, I usually chose the latter. - though I worked out many of these arguments myself, I think I can con?dently predict that few, if any, of them are novel. I also made an effort to cover some topics that seem to have been somewhat neglected in the expository literature.


Introduction to Algebraic and Abelian Functions

Introduction to Algebraic and Abelian Functions
Author: Serge Lang
Publisher: Springer Science & Business Media
Total Pages: 178
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461257409

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Introduction to Algebraic and Abelian Functions is a self-contained presentation of a fundamental subject in algebraic geometry and number theory. For this revised edition, the material on theta functions has been expanded, and the example of the Fermat curves is carried throughout the text. This volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.


Number Theory

Number Theory
Author: Helmut Koch
Publisher: American Mathematical Soc.
Total Pages: 390
Release: 2000
Genre: Mathematics
ISBN: 9780821820544

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Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of 'higher congruences' as an important element of 'arithmetic geometry'. Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem. There are a detailed exposition of the theory of Hecke $L$-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory. The book attempts as much as possible to give simple proofs. It can be used by a beginner in algebraic number theory who wishes to see some of the true power and depth of the subject. The book is suitable for two one-semester courses, with the first four chapters serving to develop the basic material. Chapters 6 through 9 could be used on their own as a second semester course.


Algebraic Numbers and Algebraic Functions

Algebraic Numbers and Algebraic Functions
Author: Emil Artin
Publisher: American Mathematical Soc.
Total Pages: 366
Release: 2005
Genre: Mathematics
ISBN: 0821840754

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Originated from the notes of a course given at Princeton University in 1950-1951, this text offers an introduction to algebraic numbers and algebraic functions. It starts with the general theory of valuation fields, proceeds to the local class field theory, and then to the theory of function fields in one variable.


Theory of Algebraic Functions of One Variable

Theory of Algebraic Functions of One Variable
Author: Richard Dedekind
Publisher: American Mathematical Soc.
Total Pages: 162
Release: 2012-07-23
Genre: Mathematics
ISBN: 0821890344

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This book is the first English translation of the classic long paper Theorie der algebraischen Functionen einer Veranderlichen (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.


Algebraic Functions

Algebraic Functions
Author: Kenkichi Iwasawa
Publisher: American Mathematical Soc.
Total Pages: 314
Release: 1993
Genre: Mathematics
ISBN: 0821819690

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This is a translation of Iwasawa's 1973 book, Theory of Algebraic Functions originally published in Japanese. Because the book treats mainly the classical part of the theory of algebraic functions, emphasizing analytic methods, it provides an excellent introduction to the subject from the classical viewpoint. Directed at graduate students, the book requires some basic knowledge of algebra, topology, and functions of a complex variable.


Algebraic Numbers and Algebraic Functions

Algebraic Numbers and Algebraic Functions
Author: P.M. Cohn
Publisher: CRC Press
Total Pages: 208
Release: 1991-09-01
Genre: Mathematics
ISBN: 9780412361906

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This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.


Algebraic Functions

Algebraic Functions
Author: Gilbert Ames Bliss
Publisher: Courier Corporation
Total Pages: 248
Release: 2004-01-01
Genre: Mathematics
ISBN: 9780486495682

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This text surveys the approaches and basic results of all three classes of algebraic functions. Starting with a treatment of single-valued analytic functions, it introduces algebraic and rational functions and proceeds to the Riemann surface of an algebraic function, integrals of rational functions, and Abel's theorem. 1933 edition