Roots 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

Without further specification, the roots of unity are complex numbers, and subsequent sections of this article will comply with this. However the defining equation of roots of unity is meaningful over any field (and even over any unital ring) $F$ , and this allows considering roots of unity in $F$ . Whichever is the field $F$ , the roots of unity in $F$ are either complex numbers, if the characteristic of $F$ is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.

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