root of unity

A root of unityMathworldPlanetmath is a number ω such that some power ωn, where n is a positive integer, equals to 1.

Specifically, if K is a field, then the nth roots of unity in K are the numbers ω in K such that ωn=1. Equivalently, they are all the roots of the polynomialPlanetmathPlanetmath Xn-1. No matter what field K is, the polynomial can never have more than n roots. Clearly 1 is an example; if n is even, then -1 will also be an example. Beyond this, the list of possibilities depends on K.

  • If K is the set of real numbers, then 1 and -1 are the only possibilities.

  • If K is the field of the complex numbersMathworldPlanetmathPlanetmath, the fundamental theorem of algebra assures us that the polynomial Xn-1 has exactly n roots (counting multiplicities). Comparing Xn-1 with its formal derivative (, nXn-1, we see that they are coprimeMathworldPlanetmathPlanetmath, and therefore all the roots of Xn-1 are distinct. That is, there exist n distinct complex numbers ω such that ωn=1.

    If ζ=e2πi/n=cos(2π/n)+isin(2π/n), then all the nth roots of unity are: ζk=e2πki/n=cos(2πk/n)+isin(2πk/n) for k=1,2,,n.

    If drawn on the complex planeMathworldPlanetmath, the nth roots of unity are the vertices of a regular n-gon centered at the origin and with a vertex at 1.

  • If K is a finite fieldMathworldPlanetmath having pa elements, for p a prime, then every nonzero element is a pa-1th root of unity (in fact this characterizes them completely; this is the role of the Frobenius operator). For other n, the answer is more complicated. For example, if n is divisible by p, the formal derivative of Xn-1 is nXn-1, which is zero since the of K is p and n is zero modulo p. So one is not guaranteed that the roots of unity will be distinct. For example, in the field of two elements, 1=-1, so there is only one square root of 1.

If an element ω is an nth root of unity but is not an mth root of unity for any 0<m<n, then ω is called a nth root of unity. For example, the number ζ defined above is a nth root of unity. If ω is a primitive nth root of unity, then all of the primitive nth roots of unity have the form ωm for some m with gcd(m,n)=1.

The roots of unity in any field have many special relationships to one another, some of which are true in general and some of which depend on the field. It is upon these relationships that the various algorithms for computing fast Fourier transforms are based.

Finally, one could ask about similar situations where K is not a field but some more general object. Here, things are much more complicated. For example, in the ring of endomorphisms of a vector spaceMathworldPlanetmath, the unipotent linear transformations are the closest analogue to roots of unity. They still form a group, but there may be many more of them than n. In a finite groupMathworldPlanetmath, every element g has a power n such that gn=1.

Title root of unity
Canonical name RootOfUnity
Date of creation 2014-11-06 15:47:15
Last modified on 2014-11-06 15:47:15
Owner alozano (2414)
Last modified by pahio (2872)
Numerical id 17
Author alozano (2872)
Entry type Definition
Classification msc 11-00
Classification msc 11-02
Related topic CyclotomicPolynomial
Related topic ExamplesOfCyclotomicPolynomials
Related topic RamanujanSum
Related topic Unity
Related topic CriterionForConstructibilityOfRegularPolygon
Related topic BinomialEquation
Defines primitive nth root of unity
Defines primitive root of unity