text: Roots of unity The th roots of unity are the  complex numbers such that , where  is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).
  The  th roots of unity are the  first powers of , that is  The th roots of unity that have this generating property are called primitive th roots of unity; they have the form  with  coprime with . The unique primitive square root of unity is  the primitive fourth roots of unity are  and
  The th roots of unity allow expressing all th roots of a complex number  as the  products of a given th roots of  with a th root of unity.
  Geometrically, the th roots of unity lie on the unit circle of the complex plane at the vertices of a regular -gon with one vertex on the real number 1.
  As the number  is the primitive th root of unity with the smallest positive argument, it is called the principal primitive th root of unity, sometimes shortened as principal th root of unity, although this terminology can be confused with the principal value of , which is 1.