Consider \begin{align} \newcommand{\F}{\mathbb F} \newcommand{\nicefrac}[2]{#1 / #2} \F_{p^2} & \cong \nicefrac{\F_{p}[U]}{(U^2 - \alpha)} \\[4pt] \F_{p^4} & \cong \nicefrac{\F_{p^2}[V]}{(V^2 - U)} \\[4pt] \F_{p^8} & \cong \nicefrac{\F_{p^4}[W]}{(W^2 - V)} \\[4pt] \F_{p^{16}} & \cong \nicefrac{\F_{p^8}[X]}{(X^2 - W)} \end{align}
where $\alpha$ is a non-square in $\F_p$. Then we have $U=V^2=W^4=X^8$ and $U^2=\alpha$. Consider $p\equiv 13\bmod 32$, therefore, $-1$ is a square and $\alpha\neq -1$.
The aim is to calculate the $p$-Frobenius for $V,$ $W,$ and $X$. Let's start with $X$
$\newcommand{\clamp}[1]{\left( #1 \right)} \exists k:\ p=32k+13\Rightarrow X^p = \clamp{X^2}^{\frac{p-1}{2}}X = W^{16k-6}X = V^{8k-3}X = U^{4k+1}VX = \alpha^{2k}UVX$
Is there anything, that could be said about $\alpha$ in that case? On the same way, is there any information, that could be generally passed by $\alpha$ if I compute $\alpha^{4k+1}$ and $\alpha^{8k+3}$?.