# nth roots of unity

• Nov 14th 2009, 10:43 PM
Sampras
nth roots of unity
We know the solutions of $\displaystyle z^{10} = 1$ geometrically form a regular decagon. It it possible to generate the solutions of $\displaystyle z^{10} = 123$ (for example) by applying some type of symmetric group action $\displaystyle G$ on the decagon?
• Nov 14th 2009, 11:03 PM
Jhevon
Quote:

Originally Posted by Sampras
We know the solutions of $\displaystyle z^{10} = 1$ geometrically form a regular decagon. It it possible to generate the solutions of $\displaystyle z^{10} = 123$ (for example) by applying some type of symmetric group action $\displaystyle G$ on the decagon?

...why would we want to do something so complicated? Is there some other goal here?
• Nov 14th 2009, 11:04 PM
Sampras
Quote:

Originally Posted by Jhevon
...why would we want to do something so complicated? Is there some other goal here?

No just wondering.
• Nov 14th 2009, 11:09 PM
Jhevon
Quote:

Originally Posted by Sampras
No just wondering.

Oh, well, I'd just interpret it as $\displaystyle \left( \frac z{\sqrt[10]{123}}\right)^{10} = 1$ and apply the first geometric interpretation, perhaps using a change of variable $\displaystyle g = \frac z{\sqrt[10]{123}}$ to make it look like $\displaystyle g^{10} = 1$.

Anyway, geometrically what would happen is that instead of your solutions lying on the circle of radius 1 in the complex plane, they will lie on the circle of radius $\displaystyle \sqrt[10]{123}$