## Factorising the Equation for Roots of Unity

The cube roots of unity are the roots of\[\omega^3 =1 \rightarrow \omega^3 -1=0\]

. There are three cube roots of unity. The sixth roots of unity are the roots of \[\omega^6-1=0\]

. There are six xis th roots of unity. All the cube roots of unity are sixth roots of unity. This means that a factor of

\[\omega^6-1=0\]

is \[ \omega^3 -1\]

. In fact, \[\omega^6-1=(\omega^3+1)(\omega^3-1)\]

.The ninth roots of unity are the roots of nbsp;

\[\omega^99-1=0\]

. There are nine ninth roots of unity.All the cube roots of unity are ninth roots of unity. This means that a factor of

\[\omega^9-1=0\]

is \[ \omega^3 -1\]

. In fact, \[\omega^9-1=(\omega^6 +\omega^3+1)(\omega^3-1)\]

.In fact, the

\[3nth\]

roots of unity are the roots of the equation \[\omega^{3n} -1=0\]

All the cube roots of unity are

\[3nth\]

roots of unity, so as before \[\omega^3 -1\]

should be a factor of \[\omega^{3n} -1=0\]

. In fact,\[\omega^{3n} -1=(\omega^{3n-3} + \omega^{3n-6} + ...+ \omega^3 + 1)( \omega^3 -1)\]

In fact, if If there are m roots of unity, the solutions of the equation

\[\omega^{n} -1=0\]

and \[m\]

is any divisor of \[n\]

then\[\begin{equation} \begin{aligned} \omega^{n} -1 &= (\omega^{n-m} + \omega^{n-2m} + ...+ \omega^m + 1)( \omega^m -1) \\ &= (\omega^{n-m} + \omega^{n-2m} + ...+ \omega^m + 1)( \omega^{m-1} + \omega^{m-2} + \omega^{m-3} + ...+ \omega + +1)(\omega-1) \end{aligned} \end{equation}\]