On the number of N-free elements with prescribed trace

Abstract

In this paper we derive a formula for the number of N-free elements over a finite field Fq with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive several explicit formulae in special cases. In addition we show that if all the prime factors of q−1 divide m, then the number of primitive elements in Fqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,m,N(c), of elements in Fqm with multiplicative order N and having trace c∈Fq.