 in mathematics, a root of unity, occasionally called a de moivre number, is any complex number that gives 1 when raised to some integer power . roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete fourier transform. in field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. any algebraically closed field has exactly  th roots of unity, if  is not divisible by the characteristic of the field. ==general definition== an th root of unity, where  is a positive integer (i.e. ), is a number  satisfying the equation :z^n = 1. traditionally,  is assumed to be a complex number, and subsequent sections of this article will comply with this usage. generally,  can be considered for any field , or even for a unital ring. in this general formulation, an th root of unity is just an element of the group of units of order . interesting cases are finite fields and modular arithmetics, for which the article root of unity modulo n contains some information. an th root of unity is  if it is not a th root of unity for some smaller : :z^k \ne 1 \qquad (k = 1, 2, 3, \dots, n-1 ). ==elementary facts== every th root of unity  is a primitive th root of unity for some  where : if  then  is a primitive first root of unity, otherwise if  then  is a primitive second (square) root of unity, otherwise, ..., and by assumption there must be a "1" at or before the th term in the sequence. if  is an th root of unity and  then . by the definition of congruence,  for some integer . but then, : z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^
1_(number) Algebraic_numbers Complex_numbers Cyclotomic_fields Polynomials