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 cāˆˆ Fq.

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Keywords Character, Finite fields, Gaussian period, Gaussian sum, Irreducible polynomial, Mersenne prime, N-free, Prescribed coefficient, Primitive, Semi-primitive, Trace, Uniform
Persistent URL dx.doi.org/10.1016/j.jnt.2015.09.008
Journal Journal of Number Theory
Citation
Tuxanidy, A. (Aleksandr), & Wang, Q. (2016). On the number of N-free elements with prescribed trace. Journal of Number Theory, 160, 536ā€“565. doi:10.1016/j.jnt.2015.09.008