Question on roots of unity

Question

This may seem absurd but what is wrong with the next reasoning about $n$th roots of unit?. For $k,l\in\mathbb Z$ such that $0 \leq k < l \leq n-1$: $$ e^{2\pi i k/n} = (e^{\pi i})^{2 k /n} \overset{\text{Euler's identity}}{=} (-1)^{2 k /n} = ((-1)^2)^{k/n} = 1^{k/n} = 1 $$ so $e^{2\pi i k /n} = e^{2\pi i l /n}$ since it is the same reasoning for $l$. Thanks for your help, I actually don't see what is wrong.

Answer 1

Here is a simpler example: $$ -1=(-1)^{\frac22}=\left((-1)^{2}\right)^{\frac12}=1^{\frac12}=1. $$ Do you see the problem here? The problem is that $1^{\frac12}$ is the solution of the equation $x^2=1$ which is not unique and you are choosing the wrong solution.
Is like we are having a function $f$ which is not 1-1 (in my example $f=x^2$), lets say that $f(a_1)=f(a_2)=r$ and $a_1\neq a_2$, and we are arguing that $$a_1=f^{-1}\left(f(a_1)\right)=f^{-1}(r)=a_2$$ which is wrong.
In your case you are using the function $f=x^n$ to deduce that $f^{-1}(1)=1^{\frac1n}=1$ which is wrong. One of the solutions of $f^{-1}(1)$ is $e^{2\pi i k/n}$ and you are choosing $f^{-1}(1)=1$.

Answer 2

There is no canonical way of defining non-integer powers of complex numbers other than the non-negative reals. So the problem lies when you want to write $z^{st}=(z^s)^t$. You are actually providing an example that such equality does not hold in general. To make the example even simpler, you could have written $$ e^{2i\pi r}=(e^{2\pi i})^r=1^r=1. $$ This of course not true, and it just shows that the relation does not hold.