Existence of Primitive Pairs with Prescribed Traces over Finite Fields

2020/04/22 by Sharma, Hariom, Sharma, R. K.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.10719

Abstract

Let F=\mathbbF_q^m, m>6, n a positive integer, and f=p/q with p, q co-prime irreducible polynomials in F[x] and deg(p) + deg(q)= n. A sufficient condition has been obtained for the existence of primitive pairs (α, f(α)) in F such that for any prescribed a, b in E=\mathbbF_q, TrF/E (α) = a and TrF/E (α^-1) = b. Further, for every positive integer n, such a pair definitely exists for large enough (q,m). The case n = 2 is dealt separately and proved that such a pair exists for all (q,m) apart from at most 64 choices.

Related

• Existence of primitive pairs with two prescribed traces over finite fields
• Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields
• Primitive normal pairs with prescribed traces over finite fields
• Primitive pairs of rational functions with prescribed traces over finite fields
• Primitive Element Pairs with One Prescribed Trace over a Finite Field