Primitive normal pairs of elements with one prescribed trace

Abstract

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ such that Tr$_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\alpha^{-1})=a$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with degree sum $n$. In particular, for $q=5^k, ~k \geq 5$ and degree sum $n=4$, we explicitly find at most 11 choices of $(q, m)$ where existence of such pairs is not guaranteed.