On the number of N-free elements with prescribed trace

In this paper we derive a formula for the number of N  -free elements over a finite field FqFq 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−1q−1 divide m  , then the number of primitive elements in FqmFqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,m,N(c)Pq,m,N(c), of elements in FqmFqm with multiplicative order N   and having trace c∈Fqc∈Fq.