Generators of finite fields with powers of trace zero and cyclotomic function fields

Abstract

Using the relation between the problem of counting irreducible polynomials over nite elds with some prescribed co- ecients to the problem of counting rational points on curves over nite elds whose function elds are subelds of cyclotomic func- tion elds, we count the number of generators of nite elds with powers of trace zero up to some point, answering a question of Z. Reichstein, and give a few other applications of this method.