Primitive elements with prescribed traces

Abstract

Given a prime power $q$ and a positive integer $n$, let ${\mathbb{F}}_{q^n}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field ${\mathbb{F}}_q$. We investigate the existence of a non-zero element $\hat{I}¾\in {\mathbb{F}}_{q^n}$ such that $\hat{I}¾+\hat{I}{¾}^{-1}$ is primitive and $T(\hat{I}¾)=a,T(\hat{I}{¾}^{-1})=b$, where $T(\hat{I}¾)$ denotes the trace of $\hat{I}¾$ in ${\mathbb{F}}_q$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n\ge 5$. A related study involves the cubic extension ${\mathbb{F}}_{q^3}$ of ${\mathbb{F}}_q$. We show that if $q\ge 8\cdot 10^{12}$ then, for any $a\in {\mathbb{F}}_q$ we can find a primitive element $\hat{I}¾\in {\mathbb{F}}_{q^3}$ such that $\hat{I}¾+\hat{I}{¾}^{-1}$ is also a primitive element of ${\mathbb{F}}_{q^3}$, and for which the trace of $\hat{I}¾$ is equal to $a$. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.

13 pages; revised version