Cyclotomic Equation

The equation

$$x^p=1,$$

where solutions $\zeta_k=e^{2\pi i k/p}$ 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 (1836) subsequently showed that this condition is not only Sufficient, but also Necessary. An ``irreducible'' cyclotomic equation is an expression of the form

$${x^p-1\over x-1} = x^{p-1}+x^{p-2}+\ldots+1=0,$$

where $p$ is Prime. Its Roots $z_i$ satisfy $\vert z_i\vert=1$.

See also Cyclotomic Polynomial, de Moivre Number, Polygon, Primitive Root of Unity

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 99-100, 1996.

Scott, C. A. ``The Binomial Equation $x^p-1=0$.'' Amer. J. Math. 8, 261-264, 1886.

Wantzel, M. L. ``Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas.'' J. Math. pures appliq. 1, 366-372, 1836.