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.

Write $\;a=r_1e^{it}\;,\;\;b=r_2e^{is}\implies\;$
$$\Bbb R\ni\frac{a^2}{b^2}\iff a^2=r_1^2e^{2it}=rr_2^2e^{2is}\;,\;\;r\in\Bbb R\iff$$
$$ (r_1^2\cos2t-rr_2^2\cos2s)+i(r_1^2\sin2t-rr_2^2\sin2s)=0\iff$$
$$\begin{cases}r=\cfrac{r_1^2\cos2t}{r_2^2\cos2s}\\{}\\
r=\cfrac{r_1^2\sin2t}{r_2^2\sin2s}\end{cases}\;\;\;\implies\tan 2s=\tan2t$$
Can you now take it from here?
