$\textbf{Theorem}$: Let $p$ be a prime. Let $q$ be a prime that doesn't divide $p - 1$, so that $\mathbb{F}_p$ does not have an element of order $q$. Let $\zeta$ be an imaginary number whose order is $q$. Then:
The polynomial $x^{q - 1} + x^{q - 2} + \cdots + x + 1$ is one with the smallest degree which $\zeta$ satisfies.
$\mathbb{F}_p(\zeta)$ is a field with $p^{q - 1}$ elements.
All elements in $\mathbb{F}_p(\zeta)$ are in the following form, and they are all distinct: $a_0 + a_1\zeta + a_2\zeta^2 + \cdots + a_{q - 2}\zeta^{q - 2}.$
My professor gave us this theorem in our elementary number theory class and when I asked him how the proof goes, he said that it can be proven using Galois Theory (which we don't know yet) but that it might be solvable using what we already know (which is a 1 semester's worth of number theory and 1-2 semesters worth of abstract algebra).
If this is provable using what I already know (elementary number theory and abstract algebra), could I get a hint on what direction to go in to prove this?