This might be a rather broad question but: are there any properties that are both necessary and sufficient for finite field elements of trace zero?
Here, I am defining the trace as the sum of Galois conjugates:
$$ \text{Tr}_{\mathbb{F}_{p^n} / \mathbb{F}_p} (x) = x + x^p + x^{p^2} + \cdots + x^{p^{n-1}}. $$
The only such characterization I have managed to find is Hilbert's Theorem 90: for $\alpha \in \mathbb{F}_{p^n}$, $\text{Tr}(\alpha) = 0$ if and only if $\alpha = \beta - \sigma (\beta)$ for $\beta \in \mathbb{F}_{p^n}$ where $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) = \langle \sigma \rangle$.
I am hoping that there are some obscure (or maybe not-so-obscure) theorems that I am unaware of. Would also very much appreciate pointers in the direction of possible references.