Number Theory in Function Fields

Contents
Preface
1    Polynomials over  Finite Fields
Exercises    ....".,....
2    Primes,  Arithmetic Functions, and the Zeta FUnction
Exercises    .,.....,."..............
3    The Reciprocity Law
Exercises
4    Dirichlet  .1.r'J<;;L.1<;"  and Primes in  an Arithmetic Progression
Exercises
5    Algebraic FUnction Fields  and  Global Function Fields
Exercises    '........................
6    Weil  Differentials and the  Canonical Class
Exercises    ...........,.."..
7    Extensions of Function Fields,  Riemann-Hurwitz: and  the  ABC Theorem
Exercises    ",."..,
8    Constant Field  Extensions
Exercises    ......... .
9    Galois Extensions    Heeke  and  Artin L-Series
Exercises    ............. .
10  Artin's Primitive Root Conjecture
Exercises    ............. .
11  T'he Behavior of the Class  Group in Constant Field  Extensions       169
Exercises    ..............................  190
12  Cyclotomic Function Fields
Exercises    ......... .
13  Drinfeld  Modules:  An  Introduction
14  S-Units, S-Class Group,  and the Corresponding L-Functions  241
Exercises    .............................   256
15  The Brumer-Stark Conjecture
Exercises    ........... .
16  The Class )lumber Formulas in Quadratic and  Cyclotomic Function Fields
Exercises    ....... ,  .  .  .  .  .  .  .  .  ,  .
17  Average Value Theorems in  Function Fields
Exercises    ............,......
Appendix:  A  Proof of the Function Field  Riemann Hypothesls  329
Bibliography  341
Author Index  353
Subject Index  355

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