Number Theory in Function Fields. Graduate Texts in Mathematics, Band 210

Seitenzahl:

358

Gewicht:

725 g

Erschienen bei:

Springer

Kurzbeschreibung

This book studies the arithmetic of polynomial rings over finite fields, A=FÄTÜ, and its relation to elementary number theory, which 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 elementary number theory, it was noticed that Z has many properties in common with A=FÄTÜ, thereby leading one to suspect that many results which hold for Z have analogues of the ring A. The first few chapters of this book illustrate 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. After presenting the needed foundational material about function fields, later chapters explore a variety of topics, including: the ABC-conjecture, Artin's conjecture on primitive roots, Drinfeld modules, the Brumer-Stark conjecture, class number formulae, and average value theorems. This book is designed for graduate students and professionals in mathematics and related fields who want to learn 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. TOC: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.

Beschreibung

This book studies the arithmetic of polynomial rings over finite fields, A=FÄTÜ, and its relation to elementary number theory, which 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 elementary number theory, it was noticed that Z has many properties in common with A=FÄTÜ, thereby leading one to suspect that many results which hold for Z have analogues of the ring A. The first few chapters of this book illustrate 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. After presenting the needed foundational material about function fields, later chapters explore a variety of topics, including: the ABC-conjecture, Artin's conjecture on primitive roots, Drinfeld modules, the Brumer-Stark conjecture, class number formulae, and average value theorems. This book is designed for graduate students and professionals in mathematics and related fields who want to learn 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.

Inhaltsverzeichnis

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.

Portrait

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.

Mehr über...

Mehr über:  Zahlentheorie, Algebraische Geometrie, Geometrie / Algebraische Geometrie, Number Theory, Algebraic Geometry, Algebraic Function Fields, Function Fields

Mehr von: 

Mehr von:  Michael Rosen, Springer