Root of unity |
In mathematics, the n th roots of unity or Abraham de Moivre numbers are complex numbers located on the unit circle. They form the vertex of a n -sided regular polygon with one vertex on 1.
=Definition=
The complex numbers z which solve
:z^n = 1 qquad (n = 1, 2, 3, cdots )
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, cdots, n - 1). (See Exponentiation#Powers_of_one)
=Primitive roots=
The n th roots of unity form under multiplication a cyclic group of order_(group theory) n . 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 Euler s totient function( 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, frac{-1 + i sqrt{3}}{2}, frac{-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} = frac{e^{2 pi i n/n} - 1}{e^{2 pi i/n} - 1} = frac{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.
=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^{-frac{1}{2}} e^{-2 pi i j k/n} From above, the columns of this matrix are orthonormal and thus the matrix is unitary matrix. In fact, this matrix is precisely the discrete Fourier transform (although normalization and sign conventions vary).
The n th roots of unity form a irreducible group 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 matrix (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 transforms.
=Cyclotomic polynomials=
The n th roots of unity are precisely the zeros of the polynomial p ( X ) = X n − 1; the primitive n th roots of unity are precisely the zeros of the n th cyclotomic polynomial : Phi_n(X) = prod_{k=1}^{phi(n)}(X-z_k);
where z 1,..., z φ( n ) are the primitive n th roots of unity. The polynomial Φ n ( X ) has integer coefficients and is irreducible over the rational number (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 : X^n - 1 = prod_{d,mid,n} Phi_d(X).;
This formula represents the factorization of the polynomial X n - 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 ) = X 2 + X + 1 :Φ4( X ) = X 2 + 1 :Φ5( X ) = X 4 + X 3 + X 2 + X + 1 :Φ6( X ) = X 2 − X + 1 In general, 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(X)=frac{X^p-1}{X-1}=sum_{k=0}^{p-1} X^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 (mathematics) 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 group isomorphism to the multiplicative group of units of the ring Z/nZ.
As the Galois group of F n /Q is abelian, this is an was published many years before Galois.
Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field — a theorem of Leopold Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.
=References=
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