Number Theory in Function Fields

ISBN-13: 9780387953359 / Angielski / Twarda / 2002 / 358 str.

Michael I. Rosen

Opis

Elementary number theory is concerned with the arithmetic properties of the ring of integers, Z, and its field of fractions, the rational numbers, Q. Early on in the development of the subject it was noticed that Z has many properties in common with A = IF T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating this by presenting, for example, analogues of the little theorems of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in an arithmetic progression. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con- sidering finite algebraic extensions K of Q, which are called algebraic num- ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K.

Recenzje

From the reviews:

MATHEMATICAL REVIEWS

"Both in the large (choice and arrangement of the material) and in the details (accuracy and completeness of proofs, quality of explanations and motivating remarks), the author did a marvelous job. His parallel treatment of topics...for both number and function fields demonstrates the strong interaction between the respective arithmetics, and allows for motivation on either side."

Bulletin of the AMS

"... Which brings us to the book by Michael Rosen. In it, one has an excellent (and, to the author's knowledge, unique) introduction to the global theory of function fields covering both the classical theory of Artin, Hasse, Weil and presenting an introduction to Drinfeld modules (in particular, the Carlitz module and its exponential). So the reader will find the basic material on function fields and their history (i.e., Weil differentials, the Riemann-Roch Theorem etc.) leading up to Bombieri's proof of the Riemann hypothesis first established by Weil. In addition one finds chapters on Artin's primitive root Conjecture for function fields, Brumer-Stark theory, the ABC Conjecture, results on class numbers and so on. Each chapter contains a list of illuminating exercises. Rosen's book is perfect for graduate students, as well as other mathematicians, fascinated by the amazing similarities between number fields and function fields."

David Goss (Ohio State University)

Spis treści

* Polynomials over Finite Fields * Primes, Arithmetic Functions, and the Zeta Function * The Reciprocity Law * Dirichlet L-series and Primes in an Arithmetic Progression * Algebraic Function Fields and Global Function Fields * Weil Differentials and the Canonical Class * Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem * Constant Field Extensions * Galois Extensions - Artin and Hecke L-functions * Artin's Primitive Root Conjecture * The Behavior of the Class Group in Constant Field Extensions * Cyclotomic Function Fields * Drinfeld Modules, An Introduction * S-Units, S-Class Group, and the Corresponding L-functions * The Brumer-Stark Conjecture * Class Number Formulas in Quadratic and Cyclotomic Function Fields * Average Value Theorems in Function Fields

O autorze

Michael Rosen is a hugely bestselling author of poetry. Michael frequently appears on radio and gives talks. He is the Children's Laureate for 2007-2009 and the winner of the Eleanor Farjeon Award for services to children's literature. Michael lives in London.

Blurb

Elementary number theory is concerned with arithmetic properties of the ring of integers. Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting, for example, analogues of the theorems of Fermat and Euler, Wilsons theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlets theorem on primes in an arithmetic progression. After presenting the required foundational material on function fields, the later chapters explore the analogy between global function fields and algebraic number fields. A variety of topics are presented, including: the ABC-conjecture, Artins conjecture on primitive roots, the Brumer-Stark conjecture, Drinfeld modules, class number formulae, and average value theorems.
The first few chapters of this book are accessible to advanced undergraduates. The later chapters are designed for graduate students and professionals in mathematics and related fields who want to learn more about the very fruitful relationship between number theory in algebraic number fields and algebraic function fields. In this book many paths are set forth for future learning and exploration.
Michael Rosen is Professor of Mathematics at Brown University, where hes been since 1962. He has published over 40 research papers and he is the co-author of A Classical Introduction to Modern Number Theory, with Kenneth Ireland. He received the Chauvenet Prize of the Mathematical Association of America in 1999 and the Philip J. Bray Teaching Award in 2001.