Primitive normal Values of rational functions with one prescribed norm and trace over 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 $q=7^k$, $m\ge 7$ and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist.

16 pages