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 [math]\displaystyle{ \alpha }[/math] satisfying the equations

[math]\displaystyle{ \begin{align} & \alpha^n = 1 \\ & \sum_{j=0}^{n-1} \alpha^{jk} = 0 \text{ for } 1 \leq k \lt n \end{align} }[/math]

In an integral domain, every primitive n-th root of unity is also a principal [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ 3 }[/math] in the ring of integers modulo [math]\displaystyle{ 26 }[/math]; while [math]\displaystyle{ 3^3 \equiv 1 \pmod{26} }[/math] and thus [math]\displaystyle{ 3 }[/math] is a cube root of unity, [math]\displaystyle{ 1 + 3 + 3^2 \equiv 13 \pmod{26} }[/math] 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