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DEFINITION


The Complex Numbers ''z'' which solve

:z^n = 1 \qquad (n = 1, 2, 3, \dots )

are called the ''n''th roots of unity.

There are ''n'' different ''n''th roots of unity .
:e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).
(See Real Powers Of Unity )


PRIMITIVE ROOTS


The ''n''th roots of unity form under multiplication a Cyclic Group of Order ''n'', and in fact these groups comprise all of the finite multiplicative subgroups of the complex numbers, except the trivial group {0}. A Generator for this cyclic group is a primitive ''n''th root of unity. The primitive ''n''th roots of unity are e^{2 \pi i k/n} where ''k'' and ''n'' are Coprime . The number of different primitive ''n''th roots of unity is given by Euler's Totient Function , \phi(n).


EXAMPLES


There is only one first root of unity, equal to 1.

The second roots of unity are +1 and -1, of which only -1 is primitive.

The third roots of unity are

:\left\{ 1, rac{-1 + i \sqrt{3}}{2}, rac{-1 - i \sqrt{3}}{2} ight\} ,
where i is the Imaginary Unit ; the latter two roots are primitive.

The fourth roots of unity are

:\left\{ 1, +i, -1, -i ight\} ,

of which +i and -i are primitive.


SUMMATION

As long as ''n'' is at least 2, the nth roots of unity add up to 0. This fact arises in many areas of mathematics and can be proved in a number of ways. One elementary proof is to apply the formula for a Geometric Series :

:\sum_{k=0}^{n-1} e^{2 \pi i k/n} = rac{e^{2 \pi i n/n} - 1}{e^{2 \pi i/n} - 1} = rac{1-1}{e^{2 \pi i/n} - 1} = 0 .

Yet another reason for the zero summation is that the roots of unity, plotted in the complex plane, form the vertices of a Regular Polygon whose Barycenter (by symmetry) lies at the origin. This summation is a special case of the Gaussian Sum .


ORTHOGONALITY

One can use the summation formula to prove an Orthogonality relationship:

:\sum_{k=0}^{n-1} e^{-2 \pi i j k/n} \cdot e^{2 \pi i j' k/n} = n \delta_{j,j'}

where \delta is the Kronecker Delta .

The nth roots of unity can be used to form an n imes n Matrix whose (j,k)th entry is
:U_{j,k}=n^{- rac{1}{2}} e^{-2 \pi i j k/n}
From above, the columns of this matrix are Orthonormal and thus the matrix is Unitary . In fact, this matrix is precisely the Discrete Fourier Transform (although normalization and sign conventions vary).

The ''n''th 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 Eigenvector s of Hermitian Matrices (for example, of a discretized one-dimensional Laplacian with periodic boundaries), from which the orthogonality property also follows (Strang, 1999).


OMEGA NOTATION


The primitive root e^{-2 \pi i /n} (or its conjugate e^{2 \pi i /n}) is often denoted \omega_n (or sometimes simply \omega), especially in the context of Discrete Fourier Transform s.


CYCLOTOMIC POLYNOMIALS

The Polynomial p(z) = z^n - 1\! zeros are precisely the ''n''th roots of unity, each with multiplicity 1.

The ''n''th cyclotomic polynomial is defined by the fact that its zeros are precisely the ''primitive'' ''n''th roots of unity, each with multiplicity 1:
:
\Phi_n(z) = \prod_{k=1}^{ arphi(n)}(z-z_k)\;

where ''z''1,...,''z''φ(''n'') are the primitive ''n''th roots of unity, and \phi(n) is Euler's Totient Function . It may be proved that the polynomial \Phi_n(z) has integer coefficients and that it 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 ''n''th root of unity is a primitive ''d''th root of unity for exactly one positive Divisor ''d'' of ''n''. This implies that
:
z^n - 1 = \prod_{d\,\mid\,n} \Phi_d(z).\;

This formula represents the factorization of the polynomial ''z''''n'' - 1 into irreducible factors, and can also be used to compute the cyclotomic polynomials. Applying Möbius Inversion to the formula gives
:\Phi_n(z)=\prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)},
where μ is the Möbius Function . The first few cyclotomic polynomials are
:\Phi_1(z) = z - 1 \!
:\Phi_2(z) = z + 1 \!
:\Phi_3(z) = z^2 + z + 1 \!
:\Phi_4(z) = z^2 + 1 \!
:\Phi_5(z) = z^4 + z^3 + z^2 + z + 1 \!
:\Phi_6(z) = z^2 - z + 1 \!
If ''p'' is a Prime Number , then all ''p''th roots of unity except 1 are primitive ''p''th roots, and we have
:
\Phi_p(z)= rac{z^p-1}{z-1}=\sum_{k=0}^{p-1} z^k

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 ''n''th root of unity to Q, one obtains the ''n''th cyclotomic field ''F''''n''. This Field contains all ''n''th roots of unity and is the Splitting Field of the ''n''th 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'''/n'''Z'''.

As the Galois group of ''F''''n''/Q is abelian, this is an '' 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.


REFERENCES