Primitive Normal Values of Rational Functions over Finite Fields

TL;DR

This paper investigates conditions under which primitive normal elements related to rational functions exist over finite fields, providing explicit bounds and criteria for their existence in various field extensions.

Contribution

It establishes a sufficient condition for the existence of primitive normal pairs linked by rational functions over finite fields and explicitly bounds exceptions for quadratic cases.

Findings

A sufficient condition for primitive normal pairs exists.

At most 55 finite fields lack such pairs in quadratic cases.

Explicit bounds for existence in field extensions.

Abstract

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a pair $(\alpha,f(\alpha))$ of primitive normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}.$ Moreover, for $q=2^k$ and rational functions $f$ with quadratic numerators and denominators, we explicitly find that there are at most $55$ finite fields $\mathbb{F}_{q^n}$ in which such a pair $(\alpha,f(\alpha))$ of primitive normal elements may not exist.