The Cyclotomic Equation

A cyclotomic equation is any equation of the form  
\[x^p - 1 = (x-1)(x^{p-1} + x^{p-2} + ... + 1) = 0\]
  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  
\[e^{2\pi i k / p}\]
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  
\[e^{- 2\pi i k / p}\]