Let $\zeta = e^{2\pi i/110}$, and set $K = \mathbb{Q}(\zeta)$. There is an $\alpha$ in $\mathbb{Q}(\zeta^{11})$ of absolute value 1, which I'm trying to find.
Consider $\sigma$, the Galois automorphism of $K$ which sends $\zeta \mapsto \zeta^{7}$. Note that $\sigma^2$ sends $\alpha$ to its complex conjugate $\bar{\alpha}$.
I have two different equations that $\alpha$ must satisfy, of the form
$-10 = A\alpha + B\alpha^\sigma + C\bar{\alpha} + D\alpha^{\sigma^3}$ $-10 = E\alpha + F\alpha^\sigma + G\bar{\alpha} + H\alpha^{\sigma^3}$
where $A,\ldots,H \in K$ are explicit known constants (none of which are zero).
Is there any way I can get my hands on $\alpha$ based on this information? I could simultaneously eliminate $\alpha^\sigma$ and $\alpha^{\sigma^3}$ if $B/D = F/H$, and then I'd have a quadratic in $\alpha$; but unfortunately $B/D \neq F/H$.