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

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

In an integral domain, every primitive n-th root of unity is also a principal $n$-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 ${\displaystyle 3^{3}\equiv 1{\pmod {26}}}$ and thus $3$ 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.

References

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