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 ${\displaystyle \alpha }$ satisfying the equations

${\displaystyle {\begin{aligned}&\alpha ^{n}=1\\&\sum _{j=0}^{n-1}\alpha ^{jk}=0{\text{ for }}1\leq k<n\end{aligned}}}$

In an integral domain, every primitive n-th root of unity is also a principal ${\displaystyle n}$-th root of unity. In any ring, if ${\displaystyle n}$ is a power of ${\displaystyle 2}$, then any ${\displaystyle n/2}$-th root of ${\displaystyle -1}$ is a principal ${\displaystyle n}$-th root of unity.

A non-example is ${\displaystyle 3}$ in the ring of integers modulo ${\displaystyle 26}$; while ${\displaystyle 3^{3}\equiv 1{\pmod {26}}}$ and thus ${\displaystyle 3}$ is a cube root of unity, ${\displaystyle 1+3+3^{2}\equiv 13{\pmod {26}}}$ 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.