Editing Root of unity (section)

{{Short description|Number with an integer power equal to 1}}
{{Use American English|date=January 2019}}
{{Use dmy dates|date=January 2020}}
{{more citations needed|date=April 2012}}
[[File:One5Root.svg|thumb|right|The 5th roots of unity (blue points) in the [[complex plane]]]]

In [[mathematics]], a '''root of unity''' is any [[complex number]] that yields 1 when [[exponentiation|raised]] to some positive [[integer]] power {{mvar|n}}. Roots of unity are used in many branches of mathematics, and are especially important in [[number theory]], the theory of [[group character]]s, and the [[discrete Fourier transform]]. It is occasionally called a '''de Moivre number''' after French mathematician [[Abraham de Moivre]].

Roots of unity can be defined in any [[field (mathematics)|field]]. If the [[characteristic of a field|characteristic]] of the field is zero, the roots are complex numbers that are also [[algebraic integer]]s. For fields with a positive characteristic, the roots belong to a [[finite field]], and, [[converse (logic)|conversely]], every nonzero element of a finite field is a root of unity. Any [[algebraically closed field]] contains exactly {{mvar|n}} {{mvar|n}}th roots of unity, except when {{mvar|n}} is a multiple of the (positive) characteristic of the field.