Are conditions equaivalent that they are roots of unity?

Question

If I have conditions that $|z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$

Is it suffice to state they are roots of unity ?

Answer 1

No. For an $n$th root of unity we must have $z_1=e^{2k\pi i/n}$ with $k\in\mathbb Z$. So if $z_1=e^i$, it is not an $n$th root of unity for any $n$ (because $\pi$ is irrational), but still $|z_1|=1$. Let $z_2=iz_1$, $z_3=-z_1$, $z_4=-iz_1$ to have four distinct nonroots of unity that add up to $0$.

Answer 2

If $z=re^{iy}$ where $r\ge0,y$ are real

$\implies|z|=r$

So, if $|z|=1,r=1\implies z=e^{iy}$

So, $z$ is $\dfrac{2\pi}y$th root of unity

Answer 3

Hint: Since we're interested in nth root of unity so, $$x^n=1$$

$$x^n-1=0$$

Now we can use product of roots $=-1$ and other conditions depending on degree of polynomial.