Root of unity

In mathematics, an nth root of unity is a complex number z such that zⁿ = 1, where n is a positive integer.

Roots of unity in the complex numbers

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:

e^{2πit / n}

for t = 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.

As long as n is at least 2, these numbers add up to 0, a simple fact that is of constant use in mathematics. It can be proved in any number of ways, for example by recognising the sum as coming from a geometric progression.

Algebraic structure

The nth 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.

Cyclotomic polynomials

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 nth cyclotomic polynomial

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). The case of prime n, which is easier than the general assertion, follows from Eisenstein's criterion.

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

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 pth roots of unity except 1 are primitive pth roots, and we have

Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, −1, or 0; the first polynomial where this occurs is Φ₁₀₅, since 105=3×5×7 is the first product of three odd primes.

Cyclotomic fields

By adjoining a primitive nth root of unity to Q, one obtains the nth cyclotomic field F_n. This field contains all nth roots of unity and is the splitting field of the nth 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, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.