Definition

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

$z^n\; =\; 1\; ,\; .$

An nth root of unity is primitive if

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

There are n different nth roots of unity:

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

One primitive nth root of unity is

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

$z^n=(e^\{2\; pi\; i/n\})^n\; =\; e^\{2\; pi\; i\}=1\; ,\; .$

$z=e^\{2\; pi\; i\; k/n\}\; qquad\; (k\; =\; 1,\; 2,\; 3,\; dots,\; n\; )\; ,\; ,$

Examples

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

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

The two primitive fourth roots of unity are

$left\{+i,\; -i\; right\}\; .$

The four primitive fifth roots of unity are

$left\{left\; .\; frac\{usqrt\{5\}-1\}\{4\}\; +\; vsqrt\{frac\{5\; +\; usqrt\{5\}\}\{8\}\}i$

The two primitive sixth roots of unity are

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

A primitive eighth root of unity is

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

See heptadecagon for the real part of a seventeenth root of unity.

Periodicity

··· , z⁻¹, z⁰, z¹, ···

··· , z ^k·(−1), z ^k·0, z^k·1, ··· (for k = 1···n),

··· , x₋₁ , x₀ , x₁ , ···

x _j = Σ_k X_k·z^k·j = X₁·z^1·j + ··· + X_n·z^n·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^i·2·π/n = cos(2·π/n) + i·sin(2·π/n)

x _j = Σ_k A_k·cos(2·π·j·k/n) + Σ_k B_k·sin(2·π·j·k/n).

Summation

$sum\_\{k=0\}^\{n-1\}\; z^k\; =\; frac\{z^n\; -\; 1\}\{z\; -\; 1\}\; =\; 0\; .$

Orthogonality

$sum\_\{k=1\}^\{n\}\; overline\{z^\{jcdot\; k\}\}\; cdot\; z^\{j\text{'}cdot\; k\}\; =\; n\; cdotdelta\_\{j,j\text{'}\}$

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

$U\_\{j,k\}=n^\{-frac\{1\}\{2\}\}cdot\; z^\{jcdot\; k\}$

$sum\_\{k=1\}^\{n\}\; overline\{U\_\{j,k\}\}\; cdot\; U\_\{k,j\text{'}\}\; =\; delta\_\{j,j\text{'}\}\; ,$

Cyclotomic polynomials

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

The nth cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive nth roots of unity, each with multiplicity 1:

$Phi\_n(z)\; =\; prod\_\{k=1\}^\{phi(n)\}(z-z\_k);$

Every nth root of unity is a primitive dth root of unity for exactly one positive divisor d of n. This implies that

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

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

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 $Phi\_\{105\}$. 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 if n has t odd prime factors then the number t-1, up to sign, occurs as a coefficient in the nth cyclotomic polynomial.

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 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. Efficient algorithms exist for calculating such expressions.

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 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. In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries), 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.

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
• Primitive root modulo n
• Dirichlet character

Notes

References

• Lang, Serge (2002). Algebra. revised 3rd edition, New York: Springer-Verlag. ISBN 0-387-95385-X.
• Milne, James S. Algebraic Number Theory. Course Notes. (1998). .
• Milne, James S. Class Field Theory. Course Notes. (1997). .
• Neukirch, Jürgen (1986). Class Field Theory. Berlin: Springer-Verlag. ISBN 3-540-15251-2.
• Washington, Lawrence C. (1997). Cyclotomic fields. 2nd edition, New York: Springer-Verlag. ISBN 0-387-94762-0.
• Derbyshire, John (2006). Unknown Quantity. Washington, D.C.: Joseph Henry Press. ISBN 0-309-09657-X.