On a question of finite extension of a prime field Let $p$ be a prime and $F$ a finite field with $p$ elements, let $K=F(\xi)$ be a finite extension of $F$ such that $[K:F]=n$.
Then $\mathcal{G}=\mathrm{Gal}(K/F)=\langle \sigma\rangle$ where $\sigma(k)=k^p$ for all $k\in K$.
Question: Is it possible that $\xi\xi^{p^m}=\xi^{p^i}$ for some $1\leq m\leq n-1$ and $1\leq i\leq n-1$?
Here, I intend to understand the conjugacy class
$$\{\sigma^j(\xi)\mid 0\leq j\leq n-1\}=\{\xi^{p^j}\mid 0\leq j\leq n-1 \}.$$
I would like to know is it possible that there are two elements in this class such that their product is still in this class.
I learned my knowledge of field theory from abstract algebra course and I knew little about trace and norm. Any explanation, references suggestion and examples are appreicated.
 A: Yes, this is possible. In the cases I see, $\xi$ needs to be a root of unity of a low order. But there may be other kinds of examples also.

The first example I thought of is when $\xi$ is a fifth root of unity, and $\Bbb{F}_2(\xi)=\Bbb{F}_{16}.$ The conjugates of $\xi$ are then (iterating Frobenius):
$$\xi^2,\ \xi^4,\ \xi^8=\xi^3,\ \text{and back to}\ \xi^{16}=\xi.$$
You see that all the non-neutral elements of $\langle\xi\rangle$ form a set of (Galois) conjugates, so there will be plenty of examples.

More generally, the same thing happens whenever $\ell$ and $p$ are prime numbers such that the residue class of $p$ generates the group $\Bbb{Z}_\ell^*$. If $\xi$ is a root of unity of order $\ell$ in an extension of $\Bbb{F}_p$, then its conjugates, $\xi^{p^i}$, $i\in\Bbb{N}$, are again the non-trivial elements of the multiplicative subgroup $\langle\xi\rangle\simeq C_\ell$. In other words, the number of conjugates of $\xi$ is equal to $\ell-1$ and $\Bbb{F}_p(\xi)=\Bbb{F}_{p^{\ell-1}}$.
The first example is an instance of this as the residue class of $p=2$ generates all of $\Bbb{Z}_\ell^*$ when $\ell=5$.

Such $(\ell,p)$ pairs are utilized in efficient computer implementations of large finite fields. Particularly in the crypto implementations of $\Bbb{F}_{2^{\ell-1}}$ when $p=2$. This is because the conjugates of $\xi$ then form what is known as an optimal normal basis. That is, a normal basis with as simple a multiplication table as possible.
