Solve Un Subseteq U2n | Roots of Unity Proof

[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.

Homework Statement

Show that U_n \subseteq U_2n for every positive integer, n.

Homework Equations

[1] U_n = {z ε ℂ, zⁿ = 1}
[2] U_n = {cos(\frac{2m\pi}{n}) + i sin(\frac{2m\pi}{n})}

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 = \pm1
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!

 

[alanlu]

Is pretty much as straightforward as you've described. Pick z in U_n = {z : zⁿ = 1}. Then you want to show z is in U_2n.

 

[HallsofIvy]

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

 

[jr16]

So, I let z be an element of the set U_n, 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 U_2n.

Is this correct?

 

[alanlu]

Yep.