Sufficient condition for existence of special type of primitive normal elements over finite fields

TL;DR

This paper establishes a sufficient condition for the existence of primitive normal elements in finite field extensions such that their quadratic transformations are also primitive normal, advancing understanding of element properties in finite fields.

Contribution

It introduces a new sufficient condition ensuring the existence of primitive normal elements with a specific quadratic form also being primitive normal over finite fields.

Findings

Provides criteria for primitive normal elements with quadratic transformations

Ensures the existence of such elements under certain conditions

Advances finite field element theory

Abstract

Let $\mathbb{F}_{q^n}$ be the extension of the field $\mathbb{F}_q$ of degree n, where $q$ is power of prime $p$, i.e $q=p^k$, where k is a positive integer. In this paper, we provide sufficient condition for the existence of a primitive normal element $\alpha\in\mathbb{F}_{q^n} $ such that $\alpha^2+\alpha+1$ is also primitive normal element over $\mathbb{F}_{q^n}$.