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

Abstract

Given $m,n,q\in \mathbb{N}$ such that $q$ is a prime power and $m\ge 3$, $a\in {\mathbb{F}}_q$, 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.