$\xi_p^i-\xi_p^j$ divides $p$ in the ring $\mathbb{Z}[\xi_p]$

Question

I am trying to prove the following:

Let $p \geq 5$ be a prime and let $\xi_p\in \mathbb{C}$ be a primitive $p$-th root of unity (that is, $\xi_p\neq 1$ and $\xi_p^p=1$). I want to show that $\xi_p^i-\xi_p^j$ properly divides $p$ in the ring $\mathbb{Z}[\xi_p]$ for any $j\neq i$.

What I have thought:

Assuming $\mathbb{Z}[\xi_p]$ were a euclidean domain (which I am not sure about), I tried to apply Euclid's algorithm and see if the remainder is zero, but did not manage to do that. I do not know if maybe a better approach would be to try to find explicitely an element of the ring which multiplied by $\xi_p^i-\xi_p^j$ gives $p$.

Any suggestions as to how to approach this problem would be very useful. Thank you.

Answer 1

Hint. Consider $\displaystyle x^{p-1}+\cdots+x+1=\frac{x^p-1}{x-1}=\prod_{\zeta\ne1}(x-\zeta)$ at $x=1$.