A primitive normal pair in a finite field with prescribed traces and norms

Abstract

Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a prime power, $t$ is a positive integer. Let $f(x)$ be a polynomial over $\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, we construct a sufficient condition on $(p,t)$ which guarantees the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=a$, $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=b$ and $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=c$, $N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=d$ where $c,d\in\mathbb{F}_{p}$ are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, we demonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only $4$ possible exceptions where such pairs may not exist.