On consecutive primitive nth roots of unity modulo q

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• 作者：Thomas Brazelton^a ; Joshua Harrington^b ; Siddarth Kannan^c ; Matthew Litman^d ; ^{myl5470@psu.edu}
• 关键词：Finite fields ; Cyclotomic polynomials ; Resultants
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：494-504
• 全文大小：290 K
• 卷排序：174

文摘

Given n∈Nn∈N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n  . In particular, we analyze the resultant of the cyclotomic polynomial Φn(x)Φn(x) with Φn(x+1)Φn(x+1), and exhibit Lucas and Mersenne divisors of this quantity. For each n≠1,2,3,6n≠1,2,3,6, we prove the existence of a prime qnqn for which there is an element α∈Zqnα∈Zqn where α   and α+1α+1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.