Principal root of unity

{{one source |date=May 2024}} In mathematics, a '''principal ''n''-th root of unity''' (where ''n'' is a positive integer) of a ring is an element <math>\alpha</math> satisfying the equations

: <math>\begin{align} & \alpha^n = 1 \\ & \sum_{j=0}^{n-1} \alpha^{jk} = 0 \text{ for } 1 \leq k < n \end{align}</math>

In an integral domain, every primitive ''n''-th root of unity is also a principal <math>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>3</math> in the ring of integers modulo <math>26</math>; while <math>3^3 \equiv 1 \pmod{26}</math> and thus <math>3</math> is a cube root of unity, <math>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== {{reflist}} *{{citation|last=Bini|first= D.|last2= Pan|first2= V.|title= Polynomial and Matrix Computations|volume=1|place= Boston, MA|publisher= Birkhäuser|year= 1994|pages=11}}

Category:Algebraic numbers Category:Cyclotomic fields Category:Polynomials Category:1 (number) Category:Complex numbers