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?



Units in cyclotomic fields
I assume you want $q$ and $r$ to be odd primes. Also, note that I will be using the notation that $\zeta\_m$ means an arbitrary primitive $m$-th root of unity (but the same one every time it appears in an equation), and will be proving the statement in that generality.


**Lemma:** For any odd $m>1$ and any $\zeta\_m$, the number $\zeta\_m+1$ is a unit.


**Proof:** Let $r$ be such that $m | 2^r-1$. We'll abbreviate $\zeta\_m$ to $\zeta$. 


Then $\zeta^{2^r} = \zeta$ so
$$1 = \left( \frac{\zeta^{2}-1}{\zeta -1} \right) \left( \frac{\zeta^{4}-1}{\zeta^{2} -1} \right) \cdots \left( \frac{\zeta^{2^r}-1}{\zeta^{2^{r-1}} -1} \right)=$$
$$\left( \zeta+1 \right) \left( \zeta^{2} + 1 \right) \cdots \left( \zeta^{2^{r-1}} +1 \right),$$
exhibiting an explicit inverse for $\zeta+1$.


Let $\eta$ be a primitive $2qr$ root of unity. Then your proposed unit is $\eta^{r}+\eta^{-r} + \eta^q + \eta^{-q}$ and factors as
$$\eta^r (1+\eta^{q-r})(1+\eta^{-q-r}).$$
Since $q$ and $r$ are odd and relatively prime, $\eta^{q-r}$ and $\eta^{q+r}$ are primitive $qr$-th roots of unity and we are done by the lemma.


