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. Roots of unity 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 a multiplicative identity element, namely a root of unity is any element of finite multiplicative order.

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

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 za = zb. 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 za 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 za = zb where 1 ≤ a < b ≤ n. Then zb−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 be a positive integer. From the above discussion, zk is a primitive root of unity for some a. Now if zka = 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:

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

i.e.

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

Therefore, zk is a primitive ath root of unity where

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

Thus, if k and n are coprime zk 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, all the roots except +1 are primitive).

In other words, if R(n) is the set of all nth roots of unity and P(n) is the set of primitive ones, R(n) is a disjoint union of the P(n):

\( \operatorname{R}(n) = \bigcup_{d\,|\,n}\operatorname{P}(d),\; \)

where the notation means that d goes through all the divisors of n, including 1 and n.

Since the cardinality of R(n) is n, and that of P(n) is φ(n), this demonstrates the classical formula

\( \sum_{d\,|\,n}\phi(d) = n.\; \)

Examples
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.

de Moivre's formula, which is valid for all real x and integers n, is

\( (\cos x + i \sin x)^n = \cos nx + i \sin nx.\, \)

Setting x = 2π/n gives a primitive nth root of unity:

\( \left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^n = \cos 2\pi + i \sin 2\pi = 1,\, \)

but for k = 1, 2, ... n−1,

\( \left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^k= \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \neq 1\; \)

This formula shows that on the complex plane the nth roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1. (See the plots for n = 3 and n = 5 on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

\( e^{i x} = \cos x + i \sin x,\, \)

which is valid for all real x, can be used to put the formula for the nth roots of unity into its most familiar form

\( e^{2\pi i \frac{k}{n}}.\; \)

It follows from the discussion in the previous section that this is a primitive root if and only if the fraction k/n is in lowest terms, i.e. that k and n are coprime.

The roots of unity are, by definition, the roots of a polynomial equation and are thus algebraic numbers. In fact, Galois theory can be used to show that they may be expressed as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots. (There are more details later in this article at Cyclotomic fields.)

The equation z1 = 1 obviously has only one solution, +1, which is therefore the only primitive first root of unity. It is a nonprimitive 2nd, 3rd, 4th, ... root of unity.

The equation z2 = 1 has two solutions, +1 and −1. +1 is the primitive first root of unity, leaving −1 as the only primitive second (square) root of unity. It is a nonprimitive 4th, 6th, 8th, ...root of unity.

The only real roots of unity are ±1; all the others are non-real complex numbers, as can be seen from de Moivre's formula or the figures.

The third (cube) roots satisfy the equation z3 − 1 = 0; the non-principal root +1 may be factored out, giving (z − 1)(z2 + z + 1) = 0. Therefore, the primitive cube roots of unity are the roots of a quadratic equation. (See Cyclotomic polynomial, below.)

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

The two primitive fourth roots of unity are the two square roots of the primitive square root of unity, −1

\left\{e^{2 \pi i/4},e^{-2 \pi i/4}\right\}=\left\{\pm\sqrt{-1} \right\}=\left\{+i, -i \right\}.\; \)

The four primitive fifth roots of unity are

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

The two primitive sixth roots of unity are the negatives (and also the square roots) of the two primitive cube roots:

\( \left\{e^{2 \pi i/6},e^{-2 \pi i/6}\right\}=\left\{ \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2} \right\} . \)

Gauss observed that if a primitive nth root of unity can be expressed using only square roots, then it is possible to construct the regular n-gon using only ruler and compass, and that if the root of unity requires third or fourth or higher radicals the regular polygon cannot be constructed. The 7th roots of unity are the first that require cube roots. Note that the real part and imaginary part are both real numbers, but complex numbers are buried in the expressions. They cannot be removed. See casus irreducibilis for details.

One of the primitive seventh roots of unity is

\( e^{2\pi i/7}=\frac{-1 + \sqrt[3]{\frac{7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{6} + \frac{i}{2}\sqrt{\frac{7 - \omega^2\sqrt[3]{\frac{7+21\sqrt{-3}}{2}} - \omega\sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{3}} \)

where ω and ω2 are the primitive cube roots of unity exp(2πi/3) and exp(4πi/3).

The four primitive eighth roots of unity are ± the square roots of the primitive fourth roots, ±i. One of them is:

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

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

\( …, z^{−1}, z^0, z^1, … \)

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

\( sk: …, z^{k·(−1)}, z^{k·0}, z^{k·1}, … \)

for k = 1, …, n are all n-periodic (because zk·(j+n) = zk·j). Furthermore, the set {s1, …, sn} 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

\( …, x_{−1}, x_0, x_1 , … \)

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

x_j = ∑_j X_j ·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 Xk is a complex amplitude.

Choosing for the primitive nth root of unity

z = e2πi/n = cos(2π/n) + i·sin(2π/n)

allows xj 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

Let SR(n) be the sum of all the nth roots of unity, primitive or not. Then

\( \operatorname{SR}(n) = \begin{cases} 1, & n=1\\ 0, & n>1. \end{cases} \)

For n = 1 there is nothing to prove. For n > 1, it is "intuitively obvious" from the symmetry of the roots in the complex plane. For a rigorous proof, let z be a primitive nth root of unity. Then the set of all roots is given by zk, k = 0, 1, ..., n−1, and their sum is given by the formula for a geometric series:

\( \sum_{k=0}^{n-1} z^k = \frac{z^n - 1}{z - 1} = 0. \)

Let SP(n) be the sum of all the primitive nth roots of unity. Then

\( \operatorname{SP}(n) = \mu(n),\; \)

where μ(n) is the Mobius function.

In the section Elementary facts, it was shown that if R(n) is the set of all nth roots of unity and P(n) is the set of primitive ones, R(n) is a disjoint union of the P(n):

\( \operatorname{R}(n) = \bigcup_{d\,|\,n}\operatorname{P}(d),\; \)

This implies

\( \operatorname{SR}(n) = \sum_{d\,|\,n}\operatorname{SP}(d).\; \)

Applying the Mobius inversion formula gives

\( \operatorname{SP}(n) = \sum_{d\,|\,n}\mu(d)\operatorname{SR}\left(\frac{n}{d}\right).\; \)

In this formula, if d < n SR(n/d) = 0, and for d = n, SR(n/d) = 1. Therefore, SP(n) = μ(n).

This is the special case cn(1) of Ramanujan's sum cn(s), defined as the sum of the sth powers of the primitive nth roots of unity:

\( c_n(s)= \sum_{a=1\atop \gcd(a,n)=1}^n e^{2 \pi i \tfrac{a}{n} s} . \)

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
Main article: Cyclotomic polynomial

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 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 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 ≥ 2 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.[1]

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 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[2] 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.[3] Efficient algorithms exist for calculating such expressions.[4]
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.[5] In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries[6]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.
Cyclotomic fields
Main article: Cyclotomic field

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.[7]

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.
See also

Circle group
Group scheme of roots of unity
Primitive root modulo n
Root of unity modulo n
Dirichlet character
Ramanujan's sum

Notes

^ 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". 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 was born in 1811, 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, ISBN 978-0-387-95385-4, MR1878556
Milne, James S. (1998). "Algebraic Number Theory". Course Notes.
Milne, James S. (1997). "Class Field Theory". Course Notes.
Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR1697859
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.