Root of unity

A root of unity is a number whose positive integer power equals 1. Its order is the least positive exponent that gives 1.

Complex plane: The plane in which complex numbers are represented by real and imaginary coordinates. Plotting roots of unity reveals their equal spacing around the unit circle.
Integer: A whole number, including zero and negative whole numbers. The positive integer exponents in the definition determine whether a number is a root of unity.
Discrete Fourier transform: A transform that decomposes a finite sequence into components at discrete frequencies. Its kernel uses powers of a primitive root of unity to encode periodic frequencies.
Root of a polynomial: A number at which a polynomial evaluates to zero. Every root of unity is a root of x^n − 1, but most polynomial roots are not roots of unity.
Unit circle: The circle of radius one centered at the origin of a coordinate plane. Every complex root of unity lies on this circle.
Divisibility: The relation in which one integer is a multiple of another with no remainder. An n-th root is also a d-th root whenever d divides n.
Fast Fourier transform: A family of algorithms that computes the discrete Fourier transform efficiently. Its factorization exploits identities among roots of unity.
Algebraic number: A complex number that is a root of a nonzero polynomial with rational coefficients. Roots of unity are algebraic numbers with the special property of finite multiplicative order.
Primitive root of unity: A root of unity whose order equals the exponent defining the group of roots under consideration. Primitive roots have the full order and generate all roots of that order.
Order of a group element: The least positive power of a group element that equals the identity, when such a power exists. The order of a root of unity is this group-theoretic order under multiplication.

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