In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers that yield 1 when raised to a given power n. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and "Moivre" redirects here for the French commune see Moivre Marne. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted They are located on the unit circle of the complex plane, and in that plane they form the vertices of an n-sided regular polygon with one vertex on 1. In Mathematics, a unit circle is In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Geometry, a vertex (plural "vertices" is a special kind of point. General properties These properties apply to both convex and star regular polygons

Definition

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.

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

see exponentiation. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Second roots are called square roots, and third roots are called cube roots. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 =  x

An nth root of unity is primitive if

There are n different nth roots of unity:

where z is any primitive nth root of unity. The primitive nth roots of unity are those z^k where k and n are coprime. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than

One primitive nth root of unity is

because

See exponentiation and Euler's identity. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

The two primitive cube roots of unity are

where i is the imaginary unit. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The two primitive fourth roots of unity are

The four primitive fifth roots of unity are

The two primitive sixth roots of unity are

A primitive eighth root of unity is

See heptadecagon for the real part of a seventeenth root of unity. Heptadecagon construction The regular heptadecagon is a Constructible polygon, as was shown by Carl Friedrich Gauss in 1796

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

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

are all n-periodic. These n sequences have furthermore the property of linear independence. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors 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 = Σ_k X_k·z^k·j = X₁·z^1·j + ··· + X_n·z^n·j . In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics

This is a form of Fourier analysis. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series If j is a (discrete) time variable, then k is a frequency and X_k is a complex amplitude. Frequency is a measure of the number of occurrences of a repeating event per unit Time. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system

Choosing for the primitive nth root of unity

z = e^i·2·π/n = cos(2·π/n) + i·sin(2·π/n)

allows x _j to be expressed as a linear combination of cos and sin

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

This is a discrete Fourier transform. In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis.

Summation

The nth roots of unity add up according to the formula for a geometric series. In Mathematics, a geometric series is a series with a constant ratio between successive terms. (This summation is a special case of the Gaussian sum. In Mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of Roots of unity, typically G (&chi &psi = ) For n > 1:

where z is a primitive nth root of unity. For n = 1, the sum has only one term (k=0) and equals 1.

Orthogonality

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

where δ is the Kronecker delta and z is any primitive nth root of unity. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

The matrix whose (j,k)th entry is

defines a discrete Fourier transform. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis. Computing the inverse transformation using gaussian elimination requires O(n³) operations. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments However, it follows from the orthogonality that U is unitary. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* That is,

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). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Mathematics, trigonometric interpolation is Interpolation with Trigonometric polynomials Interpolation is the process of finding a function which goes The straightforward application of U or its inverse to a given vector requires O(n²) operations. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments The existence of fast Fourier transform algorithms reduces the number of operations further to O(n log n).

Cyclotomic polynomials

The zeroes of the polynomial

are precisely the nth roots of unity, each with multiplicity 1. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

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

where z₁,. . . ,z_φ(n) are the primitive nth roots of unity, and φ(n) is Euler's totient function. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to The polynomial Φ_n(z) has integer coefficients and is an irreducible polynomial over the rational numbers (i. In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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. In Mathematics, Eisenstein's criterion gives sufficient conditions for a Polynomial to be irreducible over the rational

Every nth root of unity is a primitive dth root of unity for exactly one positive divisor d of n. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without This implies that

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

where μ is the Möbius function. In Mathematics, the classic Möbius inversion formula was introduced into Number theory during the 19th century by August Ferdinand Möbius. For the rational functions defined on the complex numbers see Möbius transformation.

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

. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

Substituting any positive integer for z, this sum becomes a base z repunit. In Recreational mathematics, a repunit is a Number like 11, 111, or 1111 that contains only the digit 1 Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

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 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. In Mathematics, an n th root of a Number a is a number b such that bn = a. Moreover, there exist more informative radical expressions for nth roots of unity with the additional property^[1] 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^[2]. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Efficient algorithms exist for calculating such expressions^[3]. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation

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. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak A generator for this cyclic group is a primitive nth root of unity. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

The nth roots of unity form an irreducible representation of any cyclic group of order n. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an The orthogonality relationship also follows from group-theoretic principles as described in character group. In Mathematics, a character group is the group of representations of a group by complex -valued functions.

The roots of unity appear as the eigenvectors of any circulant matrix, i. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each Row vector is rotated one element to the right relative to the preceding e. matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem. A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential ^[4] In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries^[5]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices. A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\  or \nabla^2  and named after

Cyclotomic fields

Main article: Cyclotomic field

By adjoining a primitive nth root of unity to Q, one obtains the nth cyclotomic field F_n. In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers This field contains all nth roots of unity and is the splitting field of the nth cyclotomic polynomial over Q. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, the splitting field of a Polynomial P ( X) over a given field K is a Field extension 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. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, a Galois group is a group associated with a certain type of Field extension. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in

As the Galois group of F_n/Q is abelian, this is an abelian extension. In Abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. 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. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Mathematics, in the area of Number theory, a Gaussian period is a certain kind of sum of roots of unity. The Disquisitiones Arithmeticae is a textbook of Number theory written by German Mathematician Carl Friedrich Gauss in 1798 Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German ^[6]

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. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued In Algebraic number theory, the Kronecker–Weber theorem states that every finite Abelian extension of the field of Rational numbers Q, or in

See also

• Circle group
• Primitive root modulo n
• Dirichlet character

Notes

1. ^ Landau, Susan & Miller, Gary L. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex In Modular arithmetic, a branch of Number theory, a primitive root modulo n is any number g with the property that any number Coprime In Number theory, Dirichlet characters are certain Arithmetic functions which arise from Completely multiplicative characters on the units of (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 
2. ^ Gauss, Carl F. (1965). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Disquisitiones Arithmeticae. Yale University Press, §§359–360. ISBN 0-300-09473-6.  
3. ^ 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 on 22 June 2007 
4. ^ T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1996).
5. ^ Gilbert Strang, "The discrete cosine transform," SIAM Review 41 (1), 135-147 (1999).
6. ^ The Disquisitiones was published in 1801, Galois died in 1832 but wasn't published until 1846.

References

• Lang, Serge (2002). Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. Algebra, revised 3rd edition, New York: Springer-Verlag. ISBN 0-387-95385-X.  
• Milne, James S. (1998). Algebraic Number Theory. Course Notes.
• Milne, James S. (1997). Class Field Theory. Course Notes.
• Neukirch, Jürgen (1999). Algebraic Number Theory. Berlin: Springer-Verlag. ISBN 3-540-65399-6.  
• 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.