The ?ebotarev density theorem for function fields: An elementary approach

Abstract

The Cebotarev density theorem is one of the major results in algebraic number theory. It provides a quantitative measure of sets of prime divisors with qualitative properties related to finite Galois extensions. The basic nature of the theorem makes it very handy for applications. Thus the t~ebotarev density theorem plays a central role in the decision procedure for the theory of finite fields of Ax [3] and in the transfer principle [7] from finite fields to the fields/((a), where K is a global field. An immediate application of the theorem, namely Bauer's theorem, is the key ingredient in Neukirch's proof [9], that two finite normal extensions of ~ with isomorphic absolute Galois group must coincide; and there are many more applications. The Cebotarev density theorem is true for number fields as well as for function fields of one variable over finite fields. The number field case has attracted the most attention. There are at least three versions of the proof, that of t~ebotarev [4], that of Artin [1, 2] and that of Deuring [5]. They have also found their way to textbooks, e.g., that of Lang [8]. Serre [11, Theorem7] sketches a unified treatment for both cases, via L-series The goal of this note is to provide an elementary proof for the t~ebotarev density theorem in the function field case.