\[\omega^3 =1 \rightarrow \omega^3 -1=0\]
. 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\[\begin{equation} \begin{aligned} \omega^{n} -1 &= (\omega^{n-m} + \omega^{n-2m} + ...+ \omega^m + 1)( \omega^m -1) \\ &= (\omega^{n-m} + \omega^{n-2m} + ...+ \omega^m + 1)( \omega^{m-1} + \omega^{m-2} + \omega^{m-3} + ...+ \omega + +1)(\omega-1) \end{aligned} \end{equation}\]