Number theory in function fields Machine generated contents note: 1 Polynomials over Finite Fields 1 Exercises7 2 Primes Arithmetic Functions and the Zeta Function 11 Exercises19 3 The Reciprocity Law 23 Exercises30 4 Dirilet L Series and Primes in an Arithmetic Progression 33 Exercises43 5 Algebraic Function Fields and Global Function Fields 45 Exercises59 6 Weil Differentials and the Canonical Class 63 Exercises75 7 Extensions of Function Fields Riemann Hurwitz and the ABC Theorem 77 Exercises98 8 Constant Field Extensions 101 Exercises112 9 Galois Extensions Hecke and Artin L Series 115 Exercises145 10 Artins Primitive Root Conjecture 149 Exercises166 11 The Behavior of the Class Group in Constant Field Extensions 169 Exercises190 12 Cyclotomic Function Fields 193 Exercises216 13 Drinfeld Modules: An Introduction 219 Exercises239 14 S Units S Class Group and the Corresponding L Functions 241 Exercises256 15 The Brumer Stark Conjecture 257 Exercises278 16 The Class Number Formulas in Quadratic and Cyclotomic Function Fields 283 Exercises302 17 Average Value Theorems in Function Fields 305 Exercises326 Appendix: A Proof of the Function Field Riemann Hypothesis 329 Bibliography 341 Author Index 353 Subject Index 355