mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a principal ''n''-th root of unity (where ''n'' is a positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

) of a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

is an element

\alpha

satisfying the equations :

\begin
& \alpha^n = 1 \\
& \sum_^ \alpha^ = 0 \text 1 \leq k < n
\end

In an

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...

, every primitive ''n''-th

root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...

is also a principal

n

-th root of unity. In any ring, if ''n'' is a

power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

, 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 \pmod

and thus

3

is a

cube root of unity In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...

1 + 3 + 3^2 \equiv 13 \pmod

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 In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...

to work out correctly.

References

*{{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 Algebraic numbers Cyclotomic fields Polynomials 1 (number) Complex numbers