Let $k$ be an odd rational integer, $p$ a rational prime and $\zeta_p$ a primitive $p$th root of unity. Let $\sigma$ a generator of $Gal(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, i.e., $\sigma(\zeta_p)=\zeta_p^g$ for some primitive root $g$ modolo $p$. What are the elements $z$ in $\mathbb{Q}(\zeta_p)$ with $|z|=|\sigma(z)|=\cdots=|\sigma^{p-2}(z)|$? In particular, I want to find those $z$ such that $|\sigma^r(z)|^2=kp+1$ for $r=0,1,\ldots,p-2$.