If I consider $\newcommand\F{\mathbb F}\F_{p^{16}}\cong\F_p[x]/(x^{16}-2)$. Assume that this is a valid field extension.
How to get an $A\in\F_{p^{16}}$ in isomorphic representation? Is there a nice way to choose the coefficients $A=\sum_{i=0}^{15}a_ix^i$?