2014/09/24 by Aleksandr Tuxanidy, Qiang Wang, Tuxanidy, Aleksandr +1
Computer Science · Mathematics · #11T06 #11T23 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1409.6961
openalex publication_date 2014/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we derive a formula for the number of N-free elements over a finite field \mathbbFq 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 = (qm-1)/(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 \mathbbFqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq, m, N(c), of elements in \mathbbFqm with multiplicative order N and having trace c ∈ \mathbbFq. Let N | qm-1 such that LQ | N, where LQ is the largest factor of qm-1 with the same radical as that of Q. We show there exists an element in \mathbbFqm^* of (large) order N with trace 0 if and only if m ≠ 2 and (q,m) ≠ (4,3). Moreover we derive an explicit formula for the number of elements in \mathbbFp4 with the corresponding large order LQ = 2(p+1)(p2+1) and having absolute trace zero, where p is a Mersenne prime.