Primitive Element Pairs with One Prescribed Trace over a Finite Field

2017/09/16 by Gupta, Anju, Sharma, R. K., Cohen, Stephen D.
#FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1709.05540

Abstract

In this article, we establish a sufficient condition for the existence of a primitive element α∈ \mathbbF_qⁿ such that the element α+α^-1 is also a primitive element of \mathbbF_qⁿ, and Tr_\mathbbF_qⁿ|\mathbbF_q(α)=a for any prescribed a ∈ \mathbbF_q, where q=p^k for some prime p and positive integer k. We prove that every finite field \mathbbF_qⁿ~ (n ≥5), contains such primitive elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs (q,n) for n≥5.

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