Primitive normal pairs with prescribed traces over finite fields

Abstract

Let $q$ be a positive integral power of some prime $p$ and ${\mathbb{F}}_{q^m}$ be a finite field with $q^m$ elements for some $m\in \mathbb{N}$. Here we establish a sufficient condition for the existence of primitive normal pairs of the type $(\hat{I}\mu ,f(\hat{I}\mu ))$ in ${\mathbb{F}}_{q^m}$ over ${\mathbb{F}}_q$ with two prescribed traces, $T{r}_{{\mathbb{F}}_{q^m}/{\mathbb{F}}_q}(\hat{I}\mu )=a$ and $T{r}_{{\mathbb{F}}_{q^m}/{\mathbb{F}}_q}(f(\hat{I}\mu ))=b$, where $f(x)\in {\mathbb{F}}_{q^m}(x)$ is a rational function with some restrictions and $a,b\in {\mathbb{F}}_q$. Furthermore, for $q=5^k$, $m\ge 9$ and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist.