nth roots of unity

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nth roots of unity

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Tuesday, 04 November 2008 23:34

In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. It can be shown that they are located on the unit circle of the complex plane and that in that plane they form the vertices of an n-sided regular polygon with one vertex on 1.

Definition

The complex numbers z which solve

are called the nth roots of unity. It can be shown that there are n different nth roots of unity, and that they are of the form

Primitive roots

The nth roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive nth root of unity. The primitive nth roots of unity are e2πik / n where k and n are coprime. The number of different primitive nth roots of unity is given by Euler's totient function, φ(n).
The primitive root e − 2πi / n (or its conjugate e2πi / n) is often denoted by Wn or ωn (or sometimes simply W or ω when n can be inferred from context), especially in the context of discrete Fourier transforms where this quantity occurs frequently. It is also commonly denoted ζ or ζn.

Examples

There is only one first root of unity, equal to 1.
The second roots (square roots) of unity are +1 and -1, of which only -1 is primitive.
The third roots (cubic roots) of unity are

where i is the imaginary unit; the latter two roots are primitive.
The fourth roots of unity are

of which + i and − i are primitive.
The 5th roots of unity are

A primitive 8th root of unity is

Summation

The nth roots of unity add up according to the formula for a geometric series. This summation is a special case of the Gaussian sum.
For n = 1:

For n > 1:

Orthogonality

One can use the summation formula to prove an orthogonality relationship: for j = 0, 1, ···, n − 1 and j ' = 0, 1, ···, n − 1

where δ is the Kronecker delta.
The

matrix

whose (j,k)th entry is

is unitary. This matrix is the discrete Fourier transform (although normalization and sign conventions vary).
The nth roots of unity form an irreducible representation of any cyclic group of order n. The orthogonality relationship then follows from group-theoretic principles as described in character group.
The roots of unity appear as the eigenvectors of Hermitian matrices (for example, of a discretized one-dimensional Laplacian with periodic boundaries), from which the orthogonality property also follows (Strang, 1999).

Cyclotomic polynomials

The zeroes of the polynomial

are precisely the nth roots of unity, each with multiplicity 1.
The nth cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive nth roots of unity, each with multiplicity 1:

where z1,...,zφ(n) are the primitive nth roots of unity, and φ(n) is Euler's totient function. The polynomial Φn(z) has integer coefficients and is an irreducible polynomial over the rational numbers (i.e., it cannot be written as the 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 zn - 1 into irreducible factors.
z1−1 = z−1 z2−1 = (z−1)(z+1) z3−1 = (z−1)(z2+z+1) z4−1 = (z−1)(z+1)(z2+1) z5−1 = (z−1)(z4+z3+z2+z+1) z6−1 = (z−1)(z+1)(z2+z+1)(z2−z+1) z7−1 = (z−1)(z6+z5+z4+z3+z2+z+1) Applying Möbius inversion to the formula gives

where μ is the Möbius function.
So the first few cyclotomic polynomials are
Φ1(z) = z−1 Φ2(z) = (z2−1)(z−1)−1 = z+1 Φ3(z) = (z3−1)(z−1)−1 = z2+z+1 Φ4(z) = (z4−1)(z2−1)−1 = z2+1 Φ5(z) = (z5−1)(z−1)−1 = z4+z3+z2+z+1 Φ6(z) = (z6−1)(z3−1)−1(z2−1)−1(z−1) = z2−z+1 Φ7(z) = (z7−1)(z−1)−1 = z6+z5+z4+z3+z2+z+1 If p is a prime number, then all the pth roots of unity except 1 are primitive pth roots, and we have

. Substituting any positive integer for z, this sum becomes a base z repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.
Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, −1, or 0. The first exception is Φ105, since 105 = 3×5×7 is the first product of three odd primes. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if p is prime and d | Φp(d), then either d ≡ 1 mod (p), or d ≡ 0 mod (p).
Cyclotomic polynomials are trivially solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for nth roots of unity with the additional property[1] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive nth root of unity. This was already shown by Gauss in 1797. Efficient algorithms exist for calculating such expressions].

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 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 — this is the content of a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber completed the proof.

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