root of unity

A root of unity is a number $\omega$ such that some power${\omega }^n$, where $n$ is a positive integer, equals to $1$.

Specifically, if $K$ is a field, then the $n$th roots of unity in $K$are the numbers $\omega$ in $K$ such that ${\omega }^n=1$.Equivalently, they are all the roots of the polynomial $X^n-1$. Nomatter 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 alsobe an example. Beyond this, the list of possibilities depends on $K$.

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

• If $K$ is the field of the complex numbers, the fundamentaltheorem of algebra assures us that the polynomial $X^n-1$ has exactly$n$ roots (counting multiplicities). Comparing $X^n-1$ with itsformal derivative (http://planetmath.org/derivativeofpolynomial), $n{X}^{n-1}$, we see that they are coprime, andtherefore all the roots of $X^n-1$ are distinct. That is, there exist$n$ distinct complex numbers $\omega$ such that ${\omega }^n=1$.

  If $\zeta =e^{2\pi i/n}=\cos (2\pi /n)+i\sin (2\pi /n)$, then all the$n$th roots of unity are: ${\zeta }^k=e^{2\pi ki/n}=\cos (2\pi k/n)+i\sin (2\pi k/n)$ for $k=1,2,\mathrm{\dots },n$.

  If drawn on the complex plane, the $n$th roots of unity are thevertices of a regular $n$-gon centered at the origin and with a vertexat $1$.

• If $K$ is a finite field having $p^a$ elements, for $p$ a prime,then every nonzero element is a $p^a-1$th root of unity (infact this characterizes them completely; this is the role of theFrobenius operator). For other $n$, the answer is more complicated.For example, if $n$ is divisible by $p$, the formal derivative of$X^n-1$ is $n{X}^{n-1}$, which is zero since thehttp://planetmath.org/node/1160characteristic of $K$ is $p$ and $n$ is zero modulo$p$. So one is not guaranteed that the roots of unity will bedistinct. For example, in the field of two elements, $1=-1$, so thereis only one square root of $1$.

If an element $\omega$ is an $n$th root of unity but is not an $m$throot of unity for any , then $\omega$ is called a $n$th root of unity. For example,the number $\zeta$ defined above is a $n$th root of unity. If $\omega \in ℂ$ is a primitive $n$throot of unity, then all of the primitive $n$th roots of unity have theform ${\omega }^m$ for some $m\in \mathbb{Z}$ with $\gcd (m,n)=1$.

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

Finally, one could ask about similar situations where $K$ is not afield but some more general object. Here, things are much morecomplicated. For example, in the ring of endomorphisms of a vectorspace, the unipotent linear transformations are the closest analogueto roots of unity. They still form a group, but there may be manymore of them than $n$. In a finite group, every element $g$ has apower $n$ such that $g^n=1$.