In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element α {\displaystyle \alpha } satisfying the equations
α n = 1 ∑ j = 0 n − 1 α j k = 0 for 1 ≤ k < n {\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 {\displaystyle 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 {\displaystyle 3} in the ring of integers modulo 26 {\displaystyle 26} ; while 3 3 ≡ 1 ( mod 26 ) {\displaystyle 3^{3}\equiv 1{\pmod {26}}} and thus 3 {\displaystyle 3} is a cube root of unity, 1 + 3 + 3 2 ≡ 13 ( mod 26 ) {\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.
- Bini, D.; Pan, V. (1994), Polynomial and Matrix Computations, vol. 1, Boston, MA: Birkhäuser, p. 11