Basic algebra exercise I'm stuck with this problem. I think that my difficulties are more with dealing with complex numbers then with groups, but still.
Could you please help me?
Let $\mathbb{C}^{*}$ be $\mathbb{C} \setminus \{0\}$, the multiplicative group of the complex numbers without zero. 
Let $\rho$ be the equivalence relation defined so that $a\rho b$ if $\frac{a^2}{b^2} \in \mathbb{R}$.
Describe the equivalence classes of $\rho$ as subsets of the Argand-Gauss plane.
Is $\rho$ a congruence relation compatible with the multiplication in $\mathbb{C}^{*}$?
What is the normal subgroup of $\mathbb{C}^{*}$ which corresponds to $\rho$?
Hints would be appreciated too (maybe even more than full solutions).
Thank you.

If we write $a=re^{ix}$ and $b=se^{iy}$, then $a \sim b$ if and only if $e^{2i(x-y)} \in \mathbb R$ For what "angles" does this occur?
Step 1: if $a \sim b$ is it also true that $ac \sim bc$ for any $c \in \mathbb R$?
Step 2: note that $1 \in H$ for any subgroup of $\mathbb C$. We need at least all of the $z \sim 1$. What are these?
