root of unity
Specifically, if is a field, then the th roots of unity in are the numbers in such that . Equivalently, they are all the roots of the polynomial . No matter what field is, the polynomial can never have more than roots. Clearly is an example; if is even, then will also be an example. Beyond this, the list of possibilities depends on .
If is the set of real numbers, then and are the only possibilities.
If is the field of the complex numbers, the fundamental theorem of algebra assures us that the polynomial has exactly roots (counting multiplicities). Comparing with its formal derivative (http://planetmath.org/derivativeofpolynomial), , we see that they are coprime, and therefore all the roots of are distinct. That is, there exist distinct complex numbers such that .
If , then all the th roots of unity are: for .
If is a finite field having elements, for a prime, then every nonzero element is a th root of unity (in fact this characterizes them completely; this is the role of the Frobenius operator). For other , the answer is more complicated. For example, if is divisible by , the formal derivative of is , which is zero since the http://planetmath.org/node/1160characteristic of is and is zero modulo . So one is not guaranteed that the roots of unity will be distinct. For example, in the field of two elements, , so there is only one square root of .
If an element is an th root of unity but is not an th root of unity for any , then is called a th root of unity. For example, the number defined above is a th root of unity. If is a primitive th root of unity, then all of the primitive th roots of unity have the form for some with .
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 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 . In a finite group, every element has a power such that .
|Title||root of unity|
|Date of creation||2014-11-06 15:47:15|
|Last modified on||2014-11-06 15:47:15|
|Last modified by||pahio (2872)|
|Defines||primitive th root of unity|
|Defines||primitive root of unity|