# Roots of Unity Proof

1. Feb 19, 2012

### jr16

Hey everyone! I would really appreciate some help with this problem. I have been racking my brain for hours now, and nothing seems to work/convince me.

1. The problem statement, all variables and given/known data
Show that Un $\subseteq$ U2n for every positive integer, n.

2. Relevant equations
[1] Un = {z ε ℂ, zn = 1}
[2] Un = {cos($\frac{2m\pi}{n}$) + i sin($\frac{2m\pi}{n}$)}

3. The attempt at a solution
First I started out by comparing the two sets using the first equation:
(i)zn = 1

(ii)z2n = 1
(zn)2 = 1
zn = $\sqrt{1}$
zn = $\pm$1
But I was not sure if that was enough to show one is a subset of the other

So, then I tried using the second formula
(i) $\Theta$n = $\frac{2m\pi}{n}$

(ii) $\Theta$2n = $\frac{m\pi}{n}$

I hoped I could somehow deduce that given the above theta values, one must be a subset of the other

But unfortunately, I am not sure if I am going about this proof in the right manner. I would really love any guidance you could give me. Thank you in advance!

2. Feb 19, 2012

### alanlu

Is pretty much as straightforward as you've described. Pick z in Un = {z : zn = 1}. Then you want to show z is in U2n.

3. Feb 19, 2012

### HallsofIvy

Staff Emeritus
You are going at it backwards. You want to start with $z^n= 1$ and show that $z^{2n}= 1$.

4. Feb 19, 2012

### jr16

So, I let z be an element of the set Un, where zn = 1.
Then, by squaring both sides I get:
(zn)2 = 12
z2n = 1

Therefore, z must also be an element of the set U2n.

Is this correct?

5. Feb 20, 2012

Yep.