On the number of N-free elements with prescribed trace

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• 作者：Aleksandr Tuxanidy ^{AleksandrTuxanidyTor@cmail.carleton.ca" class="auth_mail" title="E-mail the corresponding author} ; Qiang Wang ; ^{wang@math.carleton.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11T06 ; 11T24
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：536-565
• 全文大小：508 K

文摘

In this paper we derive a formula for the number of N  -free elements over a finite field F_q${\mathbb{F}}_q$ 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$q-1$ divide m  , then the number of primitive elements in F_q^m${\mathbb{F}}_{q^m}$, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, P_q,m,N(c)$P_{q,m,N}(c)$, of elements in F_q^m${\mathbb{F}}_{q^m}$ with multiplicative order N   and having trace c∈F_q$c\in {\mathbb{F}}_q$.