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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 $r=0,1,\ldots,p-2$.

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Can you explicitly describe $\sigma(z)$? – Matt Sep 22 '12 at 2:16
wait do you mean the algebraic norm given by $\prod_{g \in Gal} g(z)$? if so, that's invariant under the Galois action, so all $z$ for the first question. – uncookedfalcon Sep 22 '12 at 7:17
No, I mean by $||$ the modulus for a complex number. That is, $|z|=\sqrt{x^2+y^2}$ if $z=x+yi$. – Binzhou Xia Sep 22 '12 at 7:48
$\sigma$ is defiend by $\sigma(\zeta_p)=\zeta_p^g$ for some primitive root $g$ modolo $p$. Then $\sigma^r(\zeta_p)=\zeta_p^{g^r}$ for any integer $r$.@Matt – Binzhou Xia Sep 22 '12 at 8:03

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