In mathematics, a root of unity is any complex number that yields 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. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.
General definition
An nth root of unity, where n is a positive integer, is a number z satisfying the equation
z n = 1. {\displaystyle z^{n}=1.}
Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if n is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are
exp ( 2 k π i n ) = cos 2 k π n + i sin 2 k π n , k = 0 , 1 , … , n − 1. {\displaystyle \exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.}
However, the defining equation of roots of unity is meaningful over any field (and even over any ring) F, and this allows considering roots of unity in F. If the characteristic of F is not 0, these roots 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. An nth root of unity is said to be primitive if it is not an mth root of unity for some smaller m, that is if
z n = 1 and z m ≠ 1 for m = 1 , 2 , 3 , … , n − 1. {\displaystyle z^{n}=1\quad {\text{and}}\quad z^{m}\neq 1{\text{ for }}m=1,2,3,\ldots ,n-1.}
If n is a prime number, then all nth roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see Finite field § Roots of unity. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.
Elementary properties Every nth root of unity z is a primitive ath root of unity for some a ≤ n, which is the smallest positive integer such that za = 1. Any integer power of an nth root of unity is also an nth root of unity, as
( z k ) n = z k n = ( z n ) k = 1 k = 1. {\displaystyle {\bigl (}z^{k}{\bigr )}^{n}=z^{kn}={\bigl (}z^{n}{\bigr )}^{k}=1^{k}=1.}
This is also true for negative exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity:
1 z = z − 1 = z n − 1 = z ¯ . {\displaystyle {\frac {1}{z}}=z^{-1}=z^{n-1}={\bar {z}}.}
If z is an nth root of unity and a ≡ b (mod n) then za = zb. Indeed, by the definition of congruence modulo n, a = b + kn for some integer k, and hence
z a = z b + k n = z b z k n = z b ( z n ) k = z b 1 k = z b . {\displaystyle z^{a}=z^{b+kn}=z^{b}z^{kn}=z^{b}{\bigl (}z^{n}{\bigr )}^{k}=z^{b}1^{k}=z^{b}.}
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