# nth roots of unity

• Nov 14th 2009, 11:43 PM
Sampras
nth roots of unity
We know the solutions of $z^{10} = 1$ geometrically form a regular decagon. It it possible to generate the solutions of $z^{10} = 123$ (for example) by applying some type of symmetric group action $G$ on the decagon?
• Nov 15th 2009, 12:03 AM
Jhevon
Quote:

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

...why would we want to do something so complicated? Is there some other goal here?
• Nov 15th 2009, 12:04 AM
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 15th 2009, 12:09 AM
Jhevon
Quote:

Originally Posted by Sampras
No just wondering.

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

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 $\sqrt[10]{123}$