There is a point I don't understand in : Edward, Fermat's Last Theorem. More precisely in the last paragraph of page 173.
Let $\alpha =\exp(2i\pi/p)$, and $u=F(\alpha) \in \mathbb{Z}[\alpha]^\times$ such that $u=-\overline{u}$, where $F(X) = \sum_{i=0}^{p-1} b_i X^i$. Then this implies $F(\alpha)=-F(\alpha^{-1})$. Assume without loss of generality $b_0=0$.
Why does it implies the following : $F(\alpha) = \sum_{i=1}^{p-1} b_i(\alpha^i-\alpha^{-i})$ ?