Primitive normal pairs with prescribed traces over finite fields

TL;DR

This paper provides conditions for the existence of primitive normal pairs with prescribed traces in finite fields, and explicitly identifies exceptional fields for certain parameters.

Contribution

It establishes a sufficient condition for primitive normal pairs with prescribed traces and identifies specific fields where such pairs may not exist for particular cases.

Findings

A sufficient condition for the existence of primitive normal pairs with prescribed traces.

Explicit identification of at most 12 exceptional fields for certain parameters.

Results applicable to finite fields with q=5^k and degree sum 4 rational functions.

Abstract

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