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 +i√3) /2 and (-1 - i√3) /2. In general, the n-th roots of unity can be written as:

<math>e^{2 \pi ij/n}</math>

for j = 0, ..., n-1 (see pi and exponential function); 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

<math>

\Phi_n(X) = \prod_{k=1}^{\phi(n)}(X-z_k)\; </math> where z₁,...,z_φ(n) are the primitive n-th roots of unity. The polynomial Φ_n(X) has integer coefficients and is irreducible over the rationals (i.e., cannot be written as a product of two positive-degree polynomials with rational coefficients).

Every n-th root of unity is a primitive d-th root of unity for exactly one positive divisor d of n. This implies that

<math>

X^n - 1 = \prod_{d|n} \Phi_d(X).\; </math> This formula represents the factorization of the polynomial Xⁿ - 1 into irreducible factors and can also be used to compute the cyclotomic polynomials recursively. The first few are

Φ₁(X) = X - 1

Φ₂(X) = X + 1

Φ₃(X) = X² + X + 1

Φ₄(X) = X² + 1

Φ₅(X) = X⁴ +X³ + X² + X + 1

Φ₆(X) = X² - X + 1

In general, if p is a prime number, then all p-th roots of unity except 1 are primitive p-th roots, and we have

<math>

\Phi_p(X)=\frac{X^p-1}{X-1}=\sum_{k=0}^{p-1} X^k </math> Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, -1, or 0.

By adjoining a primitive n-th root of unity to Q, one obtains the n-th cyclotomic field F_n. This field contains all n-th roots of unity and is the splitting field of the n-th cyclotomic polynomial over Q. The field extension F_n/Q has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring Z/nZ.

As the Galois group of F_n/Q is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. In these cases Galois theory can be written out quite explicitly in terms of Gaussian periods[?]: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker[?].