In mathematics, the n^th 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 a n-sided regular polygon with one vertex on 1.

Definition

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The complex numbers z which solve

$z^n\; =\; 1\; \backslash qquad\; (n\; =\; 1,\; 2,\; 3,\; \backslash dots\; )$

are called the n^th roots of unity.

There are n different n^th roots of unity .

$e^\{2\; \backslash pi\; i\; k/n\}\; \backslash qquad\; (k\; =\; 0,\; 1,\; 2,\; \backslash dots,\; n\; -\; 1).$

Primitive roots

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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 $e^\{2\; \backslash pi\; i\; k/n\}$ where k and n are coprime. The number of different primitive n^th roots of unity is given by Euler's totient function, $\backslash phi(n)$.

Examples

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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

$\backslash left\backslash \{\; 1,\; \backslash frac\{-1\; +\; i\; \backslash sqrt\{3\}\}\{2\},\; \backslash frac\{-1\; -\; i\; \backslash sqrt\{3\}\}\{2\}\; \backslash right\backslash \}\; ,$

where $i$ is the imaginary unit; the latter two roots are primitive.

The fourth roots of unity are

$\backslash left\backslash \{\; 1,\; +i,\; -1,\; -i\; \backslash right\backslash \}\; ,$

of which $+i$ and $-i$ are primitive.

A primitive 8^th root of unity is

$\backslash sqrt\{i\}=\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}+i\backslash frac\{\backslash sqrt\{2\}\}\{2\}.$

Summation

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As long as n is at least 2, the n^th 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:

$\backslash sum\_\{k=0\}^\{n-1\}\; e^\{2\; \backslash pi\; i\; k/n\}\; =\; \backslash frac\{e^\{2\; \backslash pi\; i\; n/n\}\; -\; 1\}\{e^\{2\; \backslash pi\; i/n\}\; -\; 1\}\; =\; \backslash frac\{1-1\}\{e^\{2\; \backslash pi\; i/n\}\; -\; 1\}\; =\; 0\; .$

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

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

$\backslash sum\_\{k=0\}^\{n-1\}\; e^\{-2\; \backslash pi\; i\; j\; k/n\}\; \backslash cdot\; e^\{2\; \backslash pi\; i\; j\text{'}\; k/n\}\; =\; n\; \backslash delta\_\{j,j\text{'}\}$

where $\backslash delta$ is the Kronecker delta.

The $n$^th roots of unity can be used to form an $n\; \backslash times\; n$ matrix whose $(j,k)$^th entry is

$U\_\{j,k\}=n^\{-\backslash frac\{1\}\{2\}\}\; e^\{-2\; \backslash pi\; i\; j\; k/n\}$

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 eigenvectors 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

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The primitive root $e^\{-2\; \backslash pi\; i\; /n\}$ (or its conjugate $e^\{2\; \backslash pi\; i\; /n\}$) is often denoted $\backslash omega\_n$ (or sometimes simply $\backslash omega$), especially in the context of discrete Fourier transforms.

Cyclotomic polynomials

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The zeroes of the polynomial $p(z)\; =\; z^n\; -\; 1\backslash !$ 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 nth roots of unity, each with multiplicity 1: $\backslash Phi\_n(z)\; =\; \backslash prod\_\{k=1\}^\{\backslash varphi(n)\}(z-z\_k)\backslash ;$ where z₁,...,z_φ(n) are the primitive n^th roots of unity, and $\backslash phi(n)$ is Euler's totient function. The polynomial $\backslash Phi\_n(z)$ has integer coefficients and is an irreducible polynomial over the rational numbers (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

$z^n\; -\; 1\; =\; \backslash prod\_\{d\backslash ,\backslash mid\backslash ,n\}\; \backslash Phi\_d(z).\backslash ;$

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

$\backslash Phi\_n(z)=\backslash prod\_\{d\backslash ,\backslash mid\; n\}(z^\{n/d\}-1)^\{\backslash mu(d)\},$

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 p^th roots of unity except 1 are primitive p^th roots, and we have

$\backslash Phi\_p(z)=\backslash frac\{z^p-1\}\{z-1\}=\backslash sum\_\{k=0\}^\{p-1\}\; z^k$

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

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By adjoining a primitive n^th root of unity to Q, one obtains the n^th cyclotomic field F_n. This field contains all nth 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/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 — this is the content of a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.

References

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Field theoryAlgebraic number theory

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