Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields

TL;DR

This paper establishes conditions for the existence of primitive pairs with a prescribed trace over finite fields, extending known results and identifying nearly complete existence for specific parameters.

Contribution

It provides a new sufficient condition for primitive pairs with prescribed trace in finite fields, especially when the rational function degree is 2 and q is a power of 5.

Findings

Primitive pairs exist under the given conditions.

Almost all pairs exist for n=2 and q=5^k, except at most 20 cases.

The results generalize previous work on primitive normal pairs.

Abstract

Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(\alpha, f(\alpha))$ in $\mathbb{F}_{q^m}$ such that $\alpha$ is normal over $\mathbb{F}_q$ and $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\alpha^{-1})=a$, where $f(x)\in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$. Further, when $n=2$ and $q=5^k$ for some $k\in \mathbb{N}$, such a pair definitely exists for all $(q, m)$ apart from at most $20$ choices.