## The Cyclotomic Equation

A cyclotomic equation is any equation of the form
$x^p - 1 = (x-1)(x^{p-1} + x^{p-2} + ... + 1) = 0$
where
$p$
is a prime.
The equation
$(x^{p-1} + x^{p-2} + ... + 1) = 0$
is called an irreducible cyclotomic equation.
The solutions of this equatins are called the pth rots of unity and are equal to
$e^{2\pi i k / p}$
for
$k=0,...,p-1$
.).
They are complex numbers. Each root with an imaginary part has a mirror complex conjugate roo. The roots when plotted on an Argand diagram are symmetric in the x axis.
The roots of inity form a group under multiplication with identity 1. Each element
$e^{2\pi i k / p}$
;has inverse
$e^{- 2\pi i k / p}$
.