If $z$ is an $n$th root of unity, prove that $1/z$ is an $n$th root of unity

Question

I'm not sure if how I'm going to prove this to be correct:

Since $z$ is an $n$th root of unity, it means $z^n = 1$

For $1/z$ to be an nth root of unity, lets take it to the power of $n$,

$(1/z)^n$, and so $1/(z^n) = 1/1 = 1$, hence, $1/z$ is an $n$th root of unity ?

Another way of writing it: $(z^m)^n = (z^n)^m = 1$ for every $m \in \Bbb{Z}$. — A.P., Mar 27 '15 at 00:17

score 1 · Answer 1 · answered Mar 27 '15 at 00:12

1

Nicely done! That's exactly the most straightforward way to prove it.

answered Mar 27 '15 at 00:12

Cameron Buie

1 Answers1