The trace of 2-primitive elements of finite fields (amended version)

Abstract

Let $q$ be a prime power and $n,r$ integers such that $r\mid q^n-1$. An element of ${\mathbb{F}}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a $2$-primitive element of ${\mathbb{F}}_{q^n}$ with arbitrarily prescribed ${\mathbb{F}}_q$ trace when $n\ge 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$. A feature of this amended version is the reduction of the discussion to extensions of prime degree $n$.

This is an amended version of [6]