The trace of 2-primitive elements of finite fields
Volume 192 / 2020
Acta Arithmetica 192 (2020), 397-419
MSC: Primary 11T30; Secondary 11T06.
DOI: 10.4064/aa190307-23-5
Published online: 29 November 2019
Abstract
Let $q$ be a prime power and $n, r$ integers such that $r\,|\, q^n-1$. An element of $\mathbb F _{q^n}$ of multiplicative order $(q^n-1)/r$ is called $r$-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 $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$.
