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

## TL;DR

This paper proves the existence of 2-primitive elements with any prescribed trace in finite fields of degree at least 3, and describes trace values for degree 2, focusing on prime degree extensions.

## Contribution

It establishes the existence of 2-primitive elements with arbitrary trace in prime degree extensions of finite fields, improving previous results and simplifying the discussion.

## Key findings

- Existence of 2-primitive elements with any trace for n ≥ 3
- Explicit trace value descriptions for n=2
- Reduction of discussion to prime degree extensions

## 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 $n\geq 3$. 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 $n$.