A principal 
th
root 
of unity is a root satisfying the equations 
and
 |
for 
,
2, ..., 
.
Therefore, every primitive root of unity
of fixed degree 
over a field is a principal root of unity, although this
is not in general true over rings (Bini and Pan 1994, p. 11).
This algebraic usage is distinct from the analytic principal
root selected by a branch convention.
