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 $z=\sigma(z)=\cdots=\sigma^{p2}(z)$? In particular, I want to find those $z$ such that $\sigma^r(z)^2=kp+1$ for $r=0,1,\ldots,p2$.
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