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).
Informally, the term "principal root" is often used to refer to the root of unity having smallest positive complex argument.
