structural:summarization, structural:scholar_edition, information:classification, quality:correction
General definition

An n th 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 n th 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. 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.

An n th root of unity is said to be primitive if it is not an m th root of unity for some smaller m, that is if

If n is a prime number, then all n th roots of unity, except 1, are primitive.

In the above formula in terms of exponential and trigonometric functions, the primitive n th 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.