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 α satisfying the equations

In an integral domain, every primitive n-th root of unity is also a principal n -th root of unity. In any ring, if n is a power of 2 , then any n / 2 -th root of − 1 is a principal n -th root of unity.

A non-example is 3 in the ring of integers modulo 26 ; while 3 3 ≡ 1 ( mod 26 ) and thus 3 is a cube root of unity, 1 + 3 + 3 2 ≡ 13 ( mod 26 ) 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.