Number Theory in Function Fields

✍ Scribed by Michael Rosen (auth.)

• Publisher: Springer-Verlag New York
• Year: 2002
• Tongue: English
• Leaves: 371
• Series: Graduate Texts in Mathematics 210
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xii
Polynomials over Finite Fields....Pages 1-9
Primes, Arithmetic Functions, and the Zeta Function....Pages 11-21
The Reciprocity Law....Pages 23-31
Dirichlet L-Series and Primes in an Arithmetic Progression....Pages 33-43
Algebraic Function Fields and Global Function Fields....Pages 45-61
Weil Differentials and the Canonical Class....Pages 63-76
Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem....Pages 77-99
Constant Field Extensions....Pages 101-113
Galois Extensions — Hecke and Artin L-Series....Pages 115-147
Artin’s Primitive Root Conjecture....Pages 149-167
The Behavior of the Class Group in Constant Field Extensions....Pages 169-191
Cyclotomic Function Fields....Pages 193-217
Drinfeld Modules: An Introduction....Pages 219-239
S -Units, S -Class Group, and the Corresponding L-Functions....Pages 241-256
The Brumer-Stark Conjecture....Pages 257-281
The Class Number Formulas in Quadratic and Cyclotomic Function Fields....Pages 283-303
Average Value Theorems in Function Fields....Pages 305-327
Back Matter....Pages 329-361

✦ Subjects

Number Theory; Field Theory and Polynomials; Algebraic Geometry