Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields

2021/01/20 by Hariom Sharma, Sharma, Hariom, R. K. Sharma +1
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.08191

openalex publication_date 2021/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given m, n, q∈ ℕ such that q is a prime power and m≥ 3, a∈ \mathbbF_q, we establish a sufficient condition for the existence of primitive pair (α, f(α)) in \mathbbF_q^m such that α is normal over \mathbbF_q and Tr_\mathbbF_q^m/\mathbbF_q(α^-1)=a, where f(x)∈ \mathbbF_q^m(x) is a rational function of degree sum n. Further, when n=2 and q=5^k for some k∈ ℕ, such a pair definitely exists for all (q, m) apart from at most 20 choices.

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