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Let $q$ and $r$ be distinct prime numbers. I noticed (computing a few cases) that $\zeta_{2q} + \zeta_{2q}^{-1} + \zeta_{2r} + \zeta_{2r}^{-1}$ is a unit (in $\mathbb{Z}[\zeta_{2qr}]$, say). Is this always true? Why is that?
Let $q$ and $r$ be distinct prime numbers. I noticed (computing a few cases) that $\zeta_{2q} + \zeta_{2q}^{-1} + \zeta_{2r} + \zeta_{2r}^{-1}$ is a unit. Is this always true? Why is that?