Existence of primitive pairs with prescribed traces over finite fields
Hariom Sharma, R. K. Sharma · Communications in Algebra · 2020
Let F=Fqm, m≥7, n a positive integer, and f=p1/p2 with p1, p2 co-prime irreducible polynomials in F[x] and deg(p1)+deg(p2)=n. We obtain a sufficient condition on (q, m), which guarantees, for any prescribed a, b in E=Fq, the existence of primitive pair (α,f(α)) in F such that TrF/E(α)=a and TrF/E(α−1)=b. Further, for every positive integer n, such a pair definitely exists for large enough 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.