Some motivation: The matrix $M$ is *Butson Hadamard* if the entries of $M$ are $k^{\textrm{th}}$ roots of unity (for some $k$), and distinct pairs of rows are orthogonal under the usual Hermitian inner product. I am interested in classifying the Butson Hadamard matrices for which some power is a real scalar matrix. (This is equivalent to the corresponding unitary matrix having finite multiplicative order.)


In the case of $2\times 2$ matrices, after some reductions, the problem is equivalent to classifying
the pairs of roots of unity $(\zeta\_{1}, \zeta\_{2})$ for which


$$\Re( \zeta\_{1} ) = \sqrt{2} \Re(\zeta\_{2}) $$


This can be restated as $2\Re(\zeta\_{2})^{2} - \Re(\zeta\_{1})^{2} = 0$. The solutions we have found, up to negation and complex conjugation, are $(i, i)$, $(1, \omega\_{8})$ and $(\omega\_{8}, \omega\_{6})$.
We ran computer searches which suggest that any further solution involves a $k^{\textrm{th}}$ root of unity for some $k \geq 33$.


Q1: Is this list of solutions to the displayed equation complete? 


Q2: More generally, what techniques exist for finding all solutions to a polynomial equation in real parts of roots of unity?



Pairs of roots of unity whose real part satisfies a polynomial identity
We have
 $$ \frac{\zeta\_1+\frac{1}{\zeta\_1}}{\zeta\_2+\frac{1}{\zeta\_2}}=\sqrt{2}. $$
By applying Galois group automorphisms we can assume that
 $\zeta\_2=e^{2\pi i/n}$ for some $n$, where we can take either value
 of $\sqrt{2}$. Solving for $\zeta\_1$ gives
 $$ \sqrt{2}\zeta\_1=\zeta\_2+\frac{1}{\zeta\_2}
 \pm\sqrt{\zeta\_2^2+\frac{1}{\zeta\_2^2}}. $$
If $n>8$ then $\zeta\_2^2+\frac{1}{\zeta\_2^2}>0$. This implies that
 $\zeta\_1$ is real, a contradiction. Thus there are no further
 solutions.


