Root of unity

In mathematics, a complex number z is called an n-th root of unity if zⁿ = 1 (here, n is a positive integer).

For every positive integer n, there are n different n-th roots of unity. For example, the third roots of unity are 1, -1/2 +i√3 /2 and -1/2 - i√3 /2. In general, the n-th roots of unity can be written as:

$e^{2\pi ij \over n}$

for j = 1...n; this is a consequence of Euler's identity. Geometrically, the n-th roots of unity are located on the unit circle in the complex plane, forming the corners of a regular n-gon.

The n-th roots of unity form a group under multiplication of complex numbers. This group is cyclic. A generator of this group is called a primitive n-th root of unity. The primitive n-th roots of unity are precisely the numbers of the form exp(2πij/n) where j and n are coprime. Therefore, there are φ(n) different primitive n-th roots of unity, where φ(n) denotes Euler's phi function.

The n-th roots of unity are precisely the zeros of the polynomial p(X) = Xⁿ - 1; the primitive n-th roots of unity are precisely the zeros of the n-th cyclotomic polynomial

$\Phi _{n}(X)=\prod _{z}(X-z)\;$

where the product extends over all primitive n-th roots of unity. Φ_n(x) has integer coefficients and is irreducible over the real field (i.e., cannot be written as a product of two positive-degree polynomials with real coefficients).