Let $p$ be a prime. For $1 \leq j,k \leq p-1$, show that $$\frac{1-\zeta_p^k}{1-\zeta_p^j}$$ is invertible in the ring $\mathbb{Z}[\zeta_p]$.
First, I wanted to show that $\frac{1-\zeta_p^k}{1-\zeta_p^j} \in \mathbb{Z}[\zeta_p]$ but I did not even manage to do that. I tried to use that $1+\zeta_p + \zeta_p^2 + \cdots + \zeta_p^{p-2} + \zeta_p^{p-1} = 0$ to deduce that $$1-\zeta_p^k = \zeta_p + \zeta_p^2 + \cdots + \zeta_p^{k-1} + \zeta_p^{k+1} + \cdots + \zeta_p^{p-1}$$ and $$1-\zeta_p^j = \zeta_p + \zeta_p^2 + \cdots + \zeta_p^{j-1} + \zeta_p^{j+1} + \cdots + \zeta_p^{p-1},$$ but then I did not see how to simplify $\frac{1-\zeta_p^k}{1-\zeta_p^j}$.
