Number Theory in Function Fields

Michael Rosen

358 pages, parution le 24/01/2002

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Résumé

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, Wilson theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet 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, Artin 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.

Contents

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.

L'auteur - Michael Rosen

Michael Rosen is Chief Enterprise Architect at Genesis Development Corporation, an IONA Technologies' Company. He has over 20 years of experience in distributed computing technologies, including transaction processing, object systems, DCE, MOM, COM, and CORBA, and he coauthored Integrating CORBA and COM Applications (Wiley).

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Michael Rosen
Parution	24/01/2002
Nb. de pages	358
Format	16 x 24,3
Couverture	Relié
Poids	684g
Intérieur	Noir et Blanc
EAN13	9780387953359
ISBN13	978-0-387-95335-9