Primitive element pairs with a prescribed trace in the cubic extension of a finite field

2022/02/02 by Andrew R. Booker, Stephen D. Cohen, Booker, Andrew R. +5
Computer Science · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2202.00829

Abstract

We prove that for any prime power q∉\3,4,5\, the cubic extension \mathbbF_q³ of the finite field \mathbbF_q contains a primitive element ξ such that ξ+ξ^-1 is also primitive, and \textrmTr_\mathbbF_q³/\mathbbF_q(ξ)=a for any prescribed a∈\mathbbF_q. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree n≥3.

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