  Let $e^{2\pi i\Q}$ denote the set of roots of unity. We consider subsets
$E\subset e^{2\pi i\Q}$ that are quasi-independent or algebraically independent
(as subsets of the discrete plane). A bijective map on $e^{2\pi i\Q}$ preserves
the algebraically independent sets iff it preserves the quasi-independent sets,
and those maps are characterized. The effect on the size of quasi-independent
sets in the $n^{th}$ roots of unity $Z_n$ of increasing a prime factor of $n$
is studied.
