Root of unity

In mathematics, the n-th roots of unity or de Moivre Numbers, named after Abraham de Moivre, are complex numbers located on the unit circle. They form the vertices of a n-sided regular polygon with one vertice on 1.

Table of contents

1 Definition 2 Examples 3 Properties 4 Cyclotomic polynomials 5 Cyclotomic fields

Definition

\nFor a given n the complex numbers z which solve

are called the n-th roots of unity. There are n different n-th roots of unity. The n-th roots of unity form a cyclic group of order n under multiplication with 1 as the identity element. A generator for this cyclic group is called primitive n-th root of unity.

Examples

\nThe third roots of unity are \n:\nThe primitive third roots of unity are\n: The fourth roots of unity are \n:\nThe primitive fourth roots of unity are\n:

Properties

\nAs a consequence of Euler's identity the n-th roots of unity can be written as\n: 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. The primitive n-th roots of unity are precisely the numbers of the form exp(2πi k/n) where k 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

\nThe 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\n:\nwhere z₁,...,z_φ(n) are the primitive n-th roots of unity. The polynomial \nΦ_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\n:\nThis 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\n:Φ₁(X) = X − 1\n:Φ₂(X) = X + 1\n:Φ₃(X) = X² + X + 1\n:Φ₄(X) = X² + 1\n:Φ₅(X) = X⁴ +X³ + X² + X + 1\n:Φ₆(X) = X² - X + 1\nIn general, if p is a prime number, then all pth roots of unity except 1 are primitive pth roots, and we have\n:\nNote 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

\nBy 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. Category:Field theoryCategory:Algebraic number theory