Root of unity
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In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. They form the vertices of a n-sided regular polygon with one vertex on 1.
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Definition
For a given n the complex numbers z which solve
- <math>z^n = 1 \qquad (n \in \mathbb{N}_0)<math>
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
The third roots of unity are
- <math>\left\{ 1, \frac{-1 + i \sqrt{3}}{2}, \frac{-1 - i \sqrt{3}}{2} \right\}.\,<math>
The primitive third roots of unity are
- <math>\left\{ \frac{-1 + i \sqrt{3}}{2}, \frac{-1 - i \sqrt{3}}{2} \right\}.\,<math>
The fourth roots of unity are
- <math>\left\{ 1, +i, -1, -i \right\}.\,<math>
The primitive fourth roots of unity are
- <math>\left\{ +i, -i \right\}.\,<math>
Properties
As a consequence of Euler's identity the n-th roots of unity can be written as
- <math>e^{2 \pi i k/n} \qquad (k,n \in \mathbb{N}_0 \mbox{ and } k < n).<math>
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. These different roots of unity can be arranged to form the elements of a unitary matrix, and are thus orthogonal to each other. A detailed exposition of the orthogonality relationship is given in the article character group.
Cyclotomic polynomials
The n-th roots of unity are precisely the zeros of the polynomial p(X) = Xn − 1; the primitive nth roots of unity are precisely the zeros of the nth cyclotomic polynomial
- <math>
\Phi_n(X) = \prod_{k=1}^{\phi(n)}(X-z_k)\; <math> 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
- <math>
X^n - 1 = \prod_{d\,\mid\,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 pth roots of unity except 1 are primitive pth 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; 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, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.