Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power $n$ . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly $n$ $n$ th roots of unity if $n$ is not divisible by the characteristic of the field.

An $n$ th root of unity, where $n$ is a positive integer (i.e. $n = 1, 2, 3, \dots$ ), is a number $z$ satisfying the equation

Traditionally, $z$ is assumed to be a complex number, and subsequent sections of this article will comply with this usage. Generally, $z \in R$ can be considered for any field $R$ , or even for a unital ring. In this general formulation, an $n$ th root of unity is just an element of the group of units of order $n$ . Interesting cases are finite fields and modular arithmetics, for which the article root of unity modulo n contains some information.

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