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The th roots of unity are roots of the cyclotomic equationwhich are known as the de Moivre numbers. The notations , , and , where the value of is understood by context, are variously used to denote the th th root of unity. is always an th root of unity, but is such a root only if is even. In general, the roots of unity form a regular polygon with sides, and each vertex lies on the unit circle.

A number is an th root of unity if and a primitive th root of unity if, in addition, is the smallest integer of , ..., for which .

A solution to the cyclotomic equationThe de Moivre numbers give the coordinates in the complex plane of the polygon vertices of a regular polygon with sides and unit radius.de Moivre number231, 451, , 6

A polynomial given by(1)where are the roots of unity in given by(2)and runs over integers relatively prime to . The prime may be dropped if the product is instead taken over primitive roots of unity, so that(3)The notation is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give extensive bibliographies for cyclotomic polynomials.The cyclotomic polynomial for can also be defined as(4)where is the Möbius function and the product is taken over the divisors of (Vardi 1991, p. 225). is an integer polynomial and an irreducible polynomial with polynomial degree , where is the totient function. Cyclotomic polynomials are returned by the Wolfram Language command Cyclotomic[n, x]. The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials.The first few cyclotomic polynomials are(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)The cyclotomic..

Let be an odd prime and the cyclotomic field of th roots of unity over the rational field. Now let be the power of which divides the class number of . Then there exist integers and such thatfor all sufficiently large . For regular primes, .

A number of the formwhereis a de Moivre number and is a prime number. Unique factorizations of cyclotomic integers fail for .

where (a de Moivre number) and is a prime.

The equationwhere solutions are the roots of unity sometimes called de Moivre numbers. Gauss showed that the cyclotomic equation can be reduced to solving a series of quadratic equations whenever is a Fermat prime. Wantzel (1836) subsequently showed that this condition is not only sufficient, but also necessary. An "irreducible" cyclotomic equation is an expression of the formwhere is prime. Its roots satisfy .

A cyclotomic field is obtained by adjoining a primitive root of unity , say , to the rational numbers . Since is primitive, is also an th root of unity and contains all of the th roots of unity,(1)For example, when and , the cyclotomic field is a quadratic field(2)(3)(4)where the coefficients are contained in .The Galois group of a cyclotomic field over the rationals is the multiplicative group of , the ring of integers (mod ). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance, , where .

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