Principal root of unity

Let $n$ be a nonnegative integer. A principal $n$-th root of unity of a ring is an element $\alpha$ satisfying the equations

• $\alpha ^{n}=1$
• $\sum _{j=0}^{n-1}\alpha ^{jk}=0$ for $1\leq k<n\qquad$

Every primitive $n$-th root of unity is also a principal $n$-th root of unity. Furthermore, 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 $3^{3}\equiv 1(mod26)$ and thus $3$ is a cube root of unity, $1+3+3^{2}\equiv 13(mod26)$ 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.

See also

• Discrete_Fourier_transform_(general)