Root of unity  

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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power Template:Mvar. 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.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly Template:Mvar Template:Mvarth roots of unity, except when Template:Mvar is a multiple of the (positive) characteristic of the field.

See also

• Argand system
• Circle group, the unit complex numbers
• Cyclotomic field
• Group scheme of roots of unity
• Dirichlet character
• Ramanujan's sum
• Witt vector
• Teichmüller character