I'm trying to solve the following exercise from my algebraic number theory class:
Let $\zeta_p$ be a p-th root of unity. We want to show that for $1\leq k,j\leq p-1$, $$\frac{1-\zeta_p^k}{1-\zeta_p^j}$$ is invertible in $\mathbb{Z}[\zeta_p]$.
I tried using the cyclotomic polynomial identity to factor this differently, but I ended up with $\frac{1+\zeta_p+\cdots+\zeta_p^k}{1+\zeta_p+\cdots+\zeta_p^j}$ and it's not clear to me how to divide this cleanly to pop back in $\mathbb{Z}[\zeta_p]$. I feel like I'm missing something obvious but I don't see it.
Hint (assuming $p$ is prime): $\zeta_p^j$ is primitive $p$-th root of unity, hence there is some $l\in \Bbb Z$ with $(\zeta_p^j)^l=\zeta_p^k$. Now use the geometric sum formula to show $$\frac{1-(\zeta_p^j)^l}{1-\zeta_p^j}=\frac{1-\zeta_p^k}{1-\zeta_p^j}\in\Bbb Z[\zeta_p]$$. Similarly show $\frac{1-\zeta_p^j}{1-\zeta_p^k}\in\Bbb Z[\zeta_p]$