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Root of unity

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

Table of contents
1 Roots of unity in the complex numbers
2 Algebraic structure
3 Cyclotomic polynomials
4 Cyclotomic fields

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:

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.

Providing 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,...,zφ(n) are the primitive n-th roots of unity. The polynomial Φn(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 Fn. This field contains all nth roots of unity and is the splitting field of the nth cyclotomic polynomial over Q. The field extension Fn/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 Fn/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.