The equation x^p = 1, where solutions ζ_k = e^(2π i k/p) 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 p is a Fermat prime. Wantzel subsequently showed that this condition is not only sufficient, but also necessary. An "irreducible" cyclotomic equation is an expression of the form (x^p - 1)/(x - 1) = x^(p - 1) + x^(p - 2) + ... + 1 = 0, where p is prime. Its roots z_i satisfy left bracketing bar z_i right bracketing bar = 1.

cyclotomic polynomial | de Moivre number | polygon | primitive root of unity