Primitive normal Values of rational functions with one prescribed norm and trace over finite fields

TL;DR

This paper establishes conditions for the existence of primitive normal pairs with prescribed norm and trace in finite fields, and identifies specific exceptions for certain rational functions over particular fields.

Contribution

It provides a sufficient condition for primitive normal pairs with prescribed norm and trace, and explicitly identifies exceptions for rational functions with linear numerator and denominator over certain finite fields.

Findings

A sufficient condition for the existence of primitive normal pairs with prescribed norm and trace.

Explicit identification of at most 6 exceptional fields for certain rational functions over specific finite fields.

Extension of primitive element theory to pairs with prescribed algebraic properties in finite fields.

Abstract

Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(\alpha, f(\alpha)) \in \mathbb{F}_{q^m}$ over $\mathbb{F}$ with a prescribed primitive norm $a$ and a non-zero trace $b$ over $\mathbb{F}$ of $\alpha$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$ with some minor restrictions. Furthermore, for $q=7^k$, $m \geq 7$ and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist.