In 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 s ...

, every

primitive Primitive may refer to: Mathematics * Primitive element (field theory) * Primitive element (finite field) * Primitive cell (crystallography) * Primitive notion, axiomatic systems * Primitive polynomial (disambiguation), one of two concepts * Pr ...

''n''-th

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

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,

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 Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

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