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root of unity (Definition)

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$. 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 numbers, the fundamental theorem of algebra assures us that the polynomial $ X^n-1$ has exactly $ n$ roots (counting multiplicities). Comparing $ X^n-1$ with its formal derivative, $ nX^{n-1}$, we see that they are coprime, and therefore 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,\ldots,n$.


    If drawn on the complex plane, the $ n$th 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 field having $ p^a$ elements, for $ p$ a prime, then every nonzero element is a $ p^a-1$th 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 $ X^n-1$ is $ nX^{n-1}$, which is zero since the characteristic 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 $ \omega$ is an $ n$th root of unity but is not an $ m$th root of unity for any $ 0<m<n$, then $ \omega$ is called a primitive $ n$th root of unity. For example, the number $ \zeta$ defined above is a primitive $ n$th root of unity. If $ \omega \in \mathbb{C}$ is a primitive $ n$th root of unity, then all of the primitive $ n$th roots of unity have the form $ \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 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 space, 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 group, every element $ g$ has a power $ n$ such that $ g^n=1$.



"root of unity" is owned by alozano. [ full author list (5) | owner history (2) ]
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See Also: cyclotomic polynomial, examples of cyclotomic polynomials, Ramanujan sum, unity, criterion for constructibility of regular polygon, binomial equation

Also defines:  primitive $n$th root of unity, primitive root of unity

Attachments:
cyclotomic extension (Definition) by alozano
extracting every $n^\mathrm{th}$ term of a series (Theorem) by rm50
cyclotomic field (Definition) by Wkbj79
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Cross-references: finite group, group, linear transformations, vector space, ring of endomorphisms, object, similar, Fourier transforms, algorithms, square root, divisible, operator, prime, finite field, vertex, origin, regular, vertices, complex plane, coprime, derivative, multiplicities, fundamental theorem of algebra, complex numbers, real numbers, even, polynomial, roots, field, integer, positive
There are 64 references to this entry.

This is version 11 of root of unity, born on 2001-10-18, modified 2007-07-10.
Object id is 351, canonical name is RootOfUnity.
Accessed 12424 times total.

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AMS MSC: 11-00 (Number theory :: General reference works )
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