Basic algebra exercise

Question

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.

Answer 1

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?

Answer 2

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?