Primitive normal pairs with prescribed traces over finite fields

2024/05/29 by Nath, Shikhamoni, Mazumder, Arpan Chandra, Basnet, Dhiren Kumar
#11T23 #12E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.19068

Abstract

Let q be a positive integral power of some prime p and \mathbbF_q^m be a finite field with q^m elements for some m ∈ ℕ. Here we establish a sufficient condition for the existence of primitive normal pairs of the type (ε, f(ε)) in \mathbbF_q^m over \mathbbF_q with two prescribed traces, Tr_\mathbbF_q^m/\mathbbF_q(ε)=a and Tr_\mathbbF_q^m/\mathbbF_q(f(ε))=b, where f(x) ∈ \mathbbF_q^m(x) is a rational function with some restrictions and a, b ∈ \mathbbF_q. Furthermore, for q=5^k, m ≥ 9 and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist.

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