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 ${\mathbb{F}}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when $Q=(q^m-1)\mathrm{/}(q-1)$ is prime. More generally, for a positive integer $N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of $N$-free elements with arbitrary trace. In addition we show that if all the prime factors of $q-1$ divide $m$, then the number of primitive elements in ${\mathbb{F}}_{q^m}$, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, $P_{q,m,N}(c)$, of elements in ${\mathbb{F}}_{q^m}$ with multiplicative order $N$ and having trace $c\in {\mathbb{F}}_q$. Let $N \mid q^m-1$suchthat$L_Q \mid N$,where$L_Q$isthelargestfactorof$q^m-1$withthesameradicalasthatof$Q$\mathrm{.}Weshowthereexistsanelementin$$\mathbb{F}$_{q^m}^*$of(large)order$N$withtrace$0$ifandonlyif$m \neq 2$and$(q,m) \neq (4,3)$. Moreover we derive an explicit formula for the number of elements in ${\mathbb{F}}_{p^4}$ with the corresponding large order $L_Q=2(p+1)(p^2+1)$ and having absolute trace zero, where $p$ is a Mersenne prime.