Principal root of unity

In mathematics, a principal "n"-th root of unity (where "n" is a positive integer) of a ring is an element formula_1 satisfying the equations
In an integral domain, every primitive "n"-th root of unity is also a principal formula_3-th root of unity. In any ring, if formula_3 is a power of formula_5, then any formula_6-th root of formula_7 is a principal formula_3-th root of unity.
A non-example is formula_9 in the ring of integers modulo formula_10; while formula_11 and thus formula_9 is a cube root of unity, formula_13 meaning that it is not a "principal" cube root of unity.
The significance of a root of unity being "principal" is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.