# Thread: nth roots of unity

1. ## 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?

2. 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?

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

No just wondering.

4. 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 $\displaystyle \sqrt[10]{123}$