Angle rank of a set (reviewed)

For any set $\{\alpha_1, \alpha_2, \ldots, \alpha_s\}$ of non-zero complex numbers, the angle rank of the set is the quantity $$\dim_{\Q}\left({\rm Span}_{\Q}\left\{ \frac{ {\rm Arg}( \alpha_i)}{2\pi}\right\} \cup \{1\} \right)-1.$$ The angle rank of a polynomial is the angle rank of the set of its non-zero roots.