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 $T{r}_{{\mathbb{F}}_{p^t}/{\mathbb{F}}_p}(\epsilon )=a$, $T{r}_{{\mathbb{F}}_{p^t}/{\mathbb{F}}_p}(f(\epsilon ))=b$ 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 $a,b\in {\mathbb{F}}_p^{\ast }$. Furthermore, we demonstrate that, for $p=11^k;$ $k\ge 1,$ $m=8$ and $t\ge 15$, there are only $4$ possible exceptions where such pairs may not exist.

The paper is not much impactful and weak result