Existence of primitive pairs with two prescribed traces over finite fields

2023/01/06 by Choudhary, Aakash, Sharma, R. K.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.02381

Abstract

Given F= \mathbbF_p^t, a field with p^t elements, where p is a prime power, t≥ 7, n are positive integers and f=f₁/f₂ is a rational function, where f₁, f₂ are relatively prime, irreducible polynomials with deg(f₁) + deg(f₂) = n in F[x]. We construct a sufficient condition on (p,t) which guarantees primitive pairing (ε, f(ε)) exists in F such that Tr_\mathbbF_p^t/\mathbbF_p(ε) = a and Tr_\mathbbF_p^t/\mathbbF_p(f(ε)) = b for any prescribed a,b ∈ \mathbbF_p. Further, we demonstrate for any positive integer n, such a pair definitely exists for large t. The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p,t) except from possible 71 values of p. A result for the case n=3 is given as well.

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