Number Theory in Function Fields

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.