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

$\begin{array}{rl}& {\alpha }^n=1\\ & \sum _{j=0}^{n-1}{\alpha }^{jk}=0\text{ for }1\le k<n\end{array}$

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 $3^3\equiv 1(\mathrm{mod}26)$ and thus $3$ is a cube root of unity, $1+3+3^2\equiv 13(\mathrm{mod}26)$ 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.

References

• Bini, D.; Pan, V. (1994), Polynomial and Matrix Computations, 1, Boston, MA: Birkhäuser, pp. 11