mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

) of a

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

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, 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 setting for studying divisibilit ...

, every primitive ''n''-th

root of unity In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...

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 the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hie ...

, 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

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