Principal root of unity explained

In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element

\alpha satisfying the equations\begin{align} & \alpha ^{n} = 1 \\ & \sum ^{n -1} _{j =0} \alpha ^{jk} = 0 for 1 \leq k < n \end{align}

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} \equiv 1 \pmod{ 26 } and thus 3 is a cube root of unity, 1 + 3 + 3 ^{2} \equiv 13 \pmod{ 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.