Recently I have been reading the book in the title of this question to brush my interest in math up again in the course of getting a certificate for teaching math in middle schools.
I know some analysis stuff, but I am pretty bad at abstract algebra. Thus, I would like a few helps on resolving some issues rising up while reading the book.
Let $v(m)$ denote the index of $m$ relative to a fixed primitive root $g$, that is, $g^{v(m)} \equiv m (\text{mod $q$})$.
On pp. 18-19, it says as follows:
[Quote begins]
Let $F(\zeta) = \sum_{r = 1}^{q - 1}A_{r}\zeta^{r}$, and suppose that $F(\zeta)$ has the property that \[ F(\zeta^{m}) = F(\zeta) \] whenever $v(m) \equiv 0 (\text{mod $e$} )$. Then, by the uniqueness of representation of a polynomial, we have \[ A_{r} = A_{s} \] whenever $r \equiv sm (\text{mod $q -1$})$, and this holds for all $m$ with $v(m) \equiv 0 (\text{mod $e$})$.
Hence $A_{r}$ depends on the residue class $(\text{mod $e$})$ to which $v(r)$ belongs.
[Quote ends]
I have no idea why the modulus appearing in the quoted part is $q - 1$ (and I think this is why I have no clues at all understanding this part), because $\zeta$ is here supposed to be a $q$th root of unity, so that considering $\zeta^{q - 1}$ may not make us go anywhere, or may it?
Besides, I am not sure why the argument concerning $m$ is introduced in the first place. What for?
Your help would be really appreciated.