In mathematics, the n^th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. They form the vertices of a n-sided regular polygon with one vertex on 1. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... Abraham de Moivre (May 26, 1667 - November 27, 1754), was a French mathematician famous for de Moivres formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Illustration of a unit circle. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Definition

For a given n the complex numbers z which solve The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

are called the n^th roots of unity. There are n different n^th roots of unity.

The n^th roots of unity form a cyclic group of order n under multiplication with 1 as the identity element. A generator for this cyclic group is called primitive n^th root of unity. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

Examples

The third roots of unity are

The primitive third roots of unity are

The fourth roots of unity are

The primitive fourth roots of unity are

Properties

As a consequence of Euler's identity the n^th roots of unity can be written as In mathematics, Eulers identity is the following equation: where: is the base of the natural logarithm, is the imaginary unit, the complex number whose square is negative one, and is Archimedes constant, the ratio of the circumference of a circle to its diameter. ...

As long as n is at least 2, these numbers add up to 0, a simple fact that is of constant use in mathematics. It can be proved in any number of ways, for example by recognising the sum as coming from a geometric progression. In mathematics, a geometric progression (also inaccurately known as a geometric series, see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

The primitive n^th roots of unity are precisely the numbers of the form exp(2πi k/n) where k and n are coprime. Therefore, there are φ(n) different primitive n^th roots of unity, where φ(n) denotes Euler's phi function. These different roots of unity can be arranged to form the elements of a unitary matrix, and are thus orthogonal to each other. A detailed exposition of the orthogonality relationship is given in the article character group. Coprime - Wikipedia /**/ @import /skins-1. ... In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a character group is the group of representations of a group by complex-valued functions. ...

Cyclotomic polynomials

The n^th roots of unity are precisely the zeros of the polynomial p(X) = Xⁿ − 1; the primitive nth roots of unity are precisely the zeros of the nth cyclotomic polynomial In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

where z₁,...,z_φ(n) are the primitive n^th roots of unity. The polynomial Φ_n(X) has integer coefficients and is irreducible over the rationals (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. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, Eisensteins criterion gives sufficient conditions for a polynomial to be irreducible over Q (or equivalently, over Z). ...

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

This formula represents the factorization of the polynomial Xⁿ - 1 into irreducible factors and can also be used to compute the cyclotomic polynomials recursively. The first few are

Φ₁(X) = X − 1

Φ₂(X) = X + 1

Φ₃(X) = X² + X + 1

Φ₄(X) = X² + 1

Φ₅(X) = X⁴ + X³ + X² + X + 1

Φ₆(X) = X² − X + 1

In general, if p is a prime number, then all pth roots of unity except 1 are primitive pth roots, and we have In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

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

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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X - ai, and such that the ai generate L over K. It can be shown that such splitting fields exist... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... 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 a way that respects the given group operations. ...

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. In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, a Gaussian period is a certain kind of sum of roots of unity. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof. Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... In algebraic number theory, the Kronecker-Weber theorem states that every finite abelian extension of the field of rational numbers , or in other words every algebraic number field whose Galois group over is abelian, is a subfield of a cyclotomic field, i. ...