I am trying to prove the following:
Let $p$ be a prime. If $u$ is a unit of $\mathbb{Z[\xi_p]}$ (where $\xi_p$ is a primitive $p$-th root of unity), then $\frac{u}{\bar{u}}\neq -\xi_p^i$ for all $i$ and that $\frac{u}{\bar{u}}= \xi_p^i$ for some $i$. ($\bar{u}$ denotes the complex conjugate of $u$).
I would find it very useful to have some hint as to how to approach this problem. If it helps, in the same exercise where this problem appears, one is also asked to prove that $\alpha^p\equiv \bar{\alpha}^p \mod p\mathbb{Z[\xi_p]} $. This I managed to prove, but I fail to see how it might be related (in case it is) to my question.
Thank you in advance for your comments.