## 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 \[e^{- 2\pi i k / p}\]

.