# Number Theory in Function Fields

Springer Science & Business Media, Jan 8, 2002 - Mathematics - 358 pages
Elementary number theory 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 the subject it was noticed that Z has many properties in common with A = IF[T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating 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. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con sidering finite algebraic extensions K of Q, which are called algebraic num ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K.

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### Contents

 Primes Arithmetic Functions and the Zeta Function 11 Dirichlet LSeries and Primes in an Arithmetic Progression 33 Weil Differentials and the Canonical Class 63 Extensions of Function Fields RiemannHurwitz 77 Constant Field Extensions 101 Galois Extensions Hecke and Artin LSeries 115 Artins Primitive Root Conjecture 149 The Behavior of the Class Group in Constant Field Extensions 169
 Cyclotomic Function Fields 193 An Introduction 219 The BrumerStark Conjecture 257 The Class Number Formulas in Quadratic 283 Average Value Theorems in Function Fields 305 A Proof of the Function Field Riemann Hypothesis 329 341 Subject Index 355 Copyright

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Page xi - In fact, the number of elements in a finite field is a power of...
Page xii - Using primary decomposition prove that a finite subgroup of the multiplicative group of a field is cyclic.