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 $$\alpha$$ satisfying the equations


 * $$\begin{align}

& \alpha^n = 1 \\ & \sum_{j=0}^{n-1} \alpha^{jk} = 0 \text{ 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.