Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. They are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete Fourier transform.

The notion of root of unity also applies to any algebraic ring with 1, namely a root of unity is any element of finite multiplicative order.

1 Definition
2 Elementary facts
3 Examples
4 Periodicity
5 Summation
6 Orthogonality
7 Cyclotomic polynomials
8 Cyclic groups
9 Cyclotomic fields
10 See also
11 Notes
12 References

Definition

An nth root of unity, where n = 1,2,3,··· is a positive integer, is a complex number z satisfying the equation

$z^n = 1. \,$

An nth root of unity is primitive if it is not a kth root of unity for some smaller k:

$z^k \ne 1 \qquad (k = 1, 2, 3, \dots, n-1 ). \,$

Elementary facts

The 3rd roots of unity

Plot of z³ − 1, in which a zero is represented by the color black.

Plot of z⁵ − 1, in which a zero is represented by the color black.

Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n: if z¹ = 1 then z is a primitive first root of unity, otherwise if z² = 1 then z is a primitive second (square) root of unity, otherwise, ..., and by assumption there must be a "1" at or before the nth term in the sequence.

If z is an nth root of unity and a ≡ b (mod n) then z^a = z^b. By the definition of congruence, a = b + kn for some integer k. But then,

$z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.\;$

Therefore, given a power z^a of z, it can be assumed that 1 ≤ a ≤ n. This is often convenient.

Any integer power of an nth root of unity is also an nth root of unity:

$(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.\,$

Here k may be negative. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity:

$\frac{1}{z} = z^{-1} = z^{n-1} = \bar z.\,\,$

Let z be a primitive nth root of unity. Then the powers z, z², ... zⁿ⁻¹, zⁿ = z⁰ = 1 are all distinct. Assume the contrary, that z^a = z^b where 1 ≤ a < b ≤ n. Then z^b−a = 1. But 0 < b−a < n, which contradicts z being primitive.

Since an nth degree polynomial equation can only have n distinct roots, this implies that the powers of a primitive root z, z², ... zⁿ⁻¹, zⁿ = z⁰ = 1 are in fact all of the nth roots of unity.

From the preceding facts it follows that if z is a primitive nth root of unity:

$z^a = z^b \iff a\equiv b \pmod{ n}.\,$

If z is not primitive there is only one implication:

$a\equiv b \pmod{ n} \implies z^a = z^b.\,$

An example showing that the converse implication is false is given by:

$n = 4, \;\; z = -1, \;\; z^2 = z^4 = 1, \;\; 2 \not\equiv 4 \pmod{4}.$

Let z be a primitive nth root of unity and let k a positive integer. From the above discussion, z^k is a primitive root of unity for some a. Now if z^ka = 1 ka must be a multiple of n. The smallest number that is divisible by both n and k is their least common multiple, denoted by lcm(n, k). It is related to their greatest common divisor, gcd(n, k), by the formula:

$k\,n = \gcd(k,n)\, \operatorname{lcm}(k,n).\;$

Therefore, z^k is a primitive ath root of unity where

$a = \frac{n}{\gcd(k,n)}.\;$

Thus, if k and n are coprime z^k is also a primitive nth root of unity, and therefore there are φ(n) (where φ is Euler's totient function) distinct primitive nth roots of unity. This implies that if n is a prime number, there are n−1 primitive nth roots of unity.

The n different nth roots of unity $z^k$ (k = 1, 2, 3, ..., n) (where z is any primitive nth root of unity) are distributed evenly over the unit circle (i.e. they are the vertices of a regular n-gon with one vertex at 1), as can be seen in the plots of the 3rd and 5th roots of unity.

Examples

One primitive nth root of unity is

$e^{2 \pi i/n},\,$

because

$\left(e^{2 \pi i/n}\right)^k \,$ $= e^{2 \pi i k/n} \neq 1 \,\qquad (k = 1, 2, 3, \dots, n-1 ),\,$

and

$\left(e^{2 \pi i/n}\right)^n = e^{2 \pi i}=1,\,$

see exponentiation and Euler's identity.

The number +1 is a square root of unity because (+1)² = 1, but it is not a primitive square root of unity because (+1)¹ = 1. So +1 is only a primitive first root of unity. The number −1 is a primitive square root of unity because (−1)¹ ≠ 1 and (−1)² = 1. For n > 2, the primitive nth roots of unity are non-real complex numbers.

The two primitive cube roots of unity are

$\left\{e^{2 \pi i/3},e^{-2 \pi i/3}\right\}=\left\{ \frac{-1 + i \sqrt{3}}{2}, \frac{-1 - i \sqrt{3}}{2} \right\} ,$

where $i$ is the imaginary unit.

The two primitive fourth roots of unity are

$\left\{e^{2 \pi i/4},e^{-2 \pi i/4}\right\}=\left\{+i, -i \right\} .$

The four primitive fifth roots of unity are

$\left\{\left.e^{2 \pi i k/5}\right|k\in \{1,-1,2,-2\}\right\}=\left\{\left . \frac{u\sqrt 5-1}4+v\sqrt{\frac{5+u\sqrt 5}8}i \right |u,v \in \{-1,1\}\right\}.$

The two primitive sixth roots of unity are the negatives of the two primitive cube roots

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

One of the primitive seventh roots of unity is^[1]

$\frac13\left( \frac{1 + \sqrt[3]{\frac{-7 + 21 i \sqrt3}{2}} + \sqrt[3]{\frac{-7 -21 i \sqrt3}{2}}}{2} + i \sqrt{1-\frac{8+\sqrt[3]{\frac{-7 + 21 i \sqrt3}{2}} + \sqrt[3]{\frac{-7 -21 i \sqrt3}{2}+ \sqrt[3]{\frac{-637 - 147 i \sqrt3}{2}} +\sqrt[3]{\frac{-637+ 147 i \sqrt3}{2}}}}{2}} \right)$

A primitive eighth root of unity is

$\sqrt{i}= e^{2\pi i/8} = \frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}.$

See heptadecagon for the real part of a 17th root of unity.

Periodicity

If z is a primitive nth root of unity, then the sequence of powers

…, z⁻¹, z⁰, z¹, …

is n-periodic (because z ^j+n = z ^j·zⁿ = z ^j·1 = z ^j for all values of j), and the n sequences of powers

s_k: …, z^k·(−1), z^k·0, z^k·1, …

for k = 1, …, n are all n-periodic (because z^k·(j+n) = z^k·j). Furthermore, the set {s₁, …, s_n} of these sequences is a basis of the linear space of all n-periodic sequences. This means that any n-periodic sequence of complex numbers

…, x₋₁, x₀, x₁ , …

can be expressed as a linear combination of powers of a primitive nth root of unity:

x_j = ∑_k X_k·z^k·j = X₁·z^1·j + ··· + X_n·z^n·j

for some complex numbers X₁, …, X_n and every integer j.

This is a form of Fourier analysis. If j is a (discrete) time variable, then k is a frequency and X_k is a complex amplitude.

Choosing for the primitive nth root of unity

z = e^2πi/n = cos(2π/n) + i·sin(2π/n)

allows x_j to be expressed as a linear combination of cos and sin:

x_j = ∑_k A_k·cos(2π·j·k/n) + ∑_k B_k·sin(2π·j·k/n).

This is a discrete Fourier transform.

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:

$\sum_{k=0}^{n-1} z^k = \frac{z^n - 1}{z - 1} = 0$

where z is a primitive nth root of unity. For n = 1, the sum has only one term: z⁰=1.

Orthogonality

From the summation formula follows an orthogonality relationship: for j = 1, ···, n and j ' = 1, ···, n

$\sum_{k=1}^{n} \overline{z^{j\cdot k}} \cdot z^{j'\cdot k} = n \cdot\delta_{j,j'}$

where $\delta$ is the Kronecker delta and z is any primitive nth root of unity.

The $n \times n$ matrix $\ U$ whose $(j,k)$ th entry is

$U_{j,k}=n^{-\frac{1}{2}}\cdot z^{j\cdot k}$

defines a discrete Fourier transform. Computing the inverse transformation using gaussian elimination requires O(n³) operations. However, it follows from the orthogonality that U is unitary. That is,

$\sum_{k=1}^{n} \overline{U_{j,k}} \cdot U_{k,j'} = \delta_{j,j'} ,$

and thus the inverse of U is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation). The straightforward application of U or its inverse to a given vector requires O(n²) operations. The fast Fourier transform algorithms reduces the number of operations further to O(n log n).

Cyclotomic polynomials

The zeroes of the polynomial

$p(z) = z^n - 1\!$

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.

$\Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-z_k)\;$

where z₁,z₂,z₃,...,z_φ(n) are the primitive nth roots of unity, and φ(n) is Euler's totient function. The polynomial Φ_n(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 by applying Eisenstein's criterion to the polynomial ((z + 1)ⁿ−1) / ((z + 1) − 1), and expanding via the binomial theorem.

Every nth root of unity is a primitive dth 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 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

$\Phi_n(z) = \prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)} = \prod_{d\,\mid n}(z^{d}-1)^{\mu(n/d)},$

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 the pth roots of unity except 1 are primitive pth roots, and we have

$\Phi_p(z)=\frac{z^p-1}{z-1}=\sum_{k=0}^{p-1} z^k.$

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 0, 1, or −1. The first exception is Φ₁₀₅. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on n as on how many odd prime factors appear in n. More precisely, it can be shown that if n has 1 or 2 odd prime factors (e.g., n = 150) then the nth cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable n for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is 3·5·7 = 105. This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if n = p₁·p₂· ... ·p_t, where p₁ < p₂ < ... < p_t are odd primes, p₁ + p₂ > p_t, and t is odd, then 1 − t occurs as a coefficient in the nth cyclotomic polynomial.^[2]

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if p is prime and d | Φ_p(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^[3] 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.^[4] Efficient algorithms exist for calculating such expressions.^[5]

Cyclic groups

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 nth roots of unity form an irreducible representation of any cyclic group of order n. The orthogonality relationship also follows from group-theoretic principles as described in character group.

The roots of unity appear as entries of the eigenvectors of any circulant matrix, i.e. matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem.^[6] In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries^[7]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

Cyclotomic fields

By adjoining a primitive nth root of unity to Q, one obtains the nth cyclotomic field F_n. This field contains all nth roots of unity and is the splitting field of the nth 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 explicitly in terms of Gaussian periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.^[8]

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.

Notes

↑ Blowers, James. "The Seventh Roots of Unity". http://www.mindspring.com/~jimvb/TheSeventhRootofUnity.pdf. Retrieved 2010-08-22
↑ Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bulletin of the American Mathematical Society 42 (1936), no. 6, pp. 389–392.
↑ Landau, Susan; Miller, Gary L. (1985). "Solvability by radicals is in polynomial time". Journal of Computer and System Sciences 30 (2): 179–208. doi:10.1016/0022-0000(85)90013-3
↑ Gauss, Carl F. (1965). Disquisitiones Arithmeticae. Yale University Press. pp. §§359–360. ISBN 0-300-09473-6.
↑ Weber, Andreas; Keckeisen, Michael. "Solving Cyclotomic Polynomials by Radical Expressions". http://cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/WeberKeckeisen99a.pdf. Retrieved 2007-06-22
↑ T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1996).
↑ Gilbert Strang, "The discrete cosine transform," SIAM Review 41 (1), 135–147 (1999).
↑ The Disquisitiones was published in 1801, Galois died in 1832 but wasn't published until 1846.

References

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR 1878556, ISBN 978-0-387-95385-4
Milne, James S. (1998). "Algebraic Number Theory". Course Notes. http://www.jmilne.org/math.
Milne, James S. (1997). "Class Field Theory". Course Notes. http://www.jmilne.org/math.
Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, MR 1697859, ISBN 978-3-540-65399-8
Neukirch, Jürgen (1986). Class Field Theory. Berlin: Springer-Verlag. ISBN 3-540-15251-2.
Washington, Lawrence C. (1997). Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0.
Derbyshire, John (2006). "Roots of Unity". Unknown Quantity. Washington, D.C.: Joseph Henry Press. ISBN 0-309-09657-X. http://www.JohnDerbyshire.com/Books/Unknown/page.html.