Proving that a polynomial is the generating idempotent of the repetition code of length $n$

Question

First of all, we consider a finite field $\mathbb{F}$, with $|\mathbb{F}|=q$, and a natural number $n$ that is relatively prime to $q$.
We also consider the quotient ring $\mathcal{R}_n = \mathbb{F}[x]/<x^n-1>$.

I want to prove that the polynomial $g(x)=\frac{1}{n}(x^{n-1}+x^{n-2}+...+x+1)$ is a generating idempotent of the repetition code of length $n$ over $\mathbb{F}$.

So far, I have managed to prove that $g(x)$ is an idempotent element of the ring $\mathcal{R}_n$.
I also know that $x^n-1 = n(x-1)g(x)$.
Any help would be appreciated.