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

TL;DR

This paper establishes conditions under which primitive normal pairs with prescribed traces and norms exist in finite fields, providing explicit exceptions for certain parameters, advancing understanding of finite field element distributions.

Contribution

It introduces new sufficient conditions on (p,t) for the existence of primitive normal pairs with prescribed traces and norms, including explicit exceptions for specific parameters.

Findings

Identifies conditions guaranteeing primitive normal pairs with prescribed properties.

Shows only 4 possible exceptions for p=11^k, m=8, t≥15.

Provides a framework for constructing such pairs in finite fields.

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.