Is there any particular study on the following group

Question

The set of all principal fractional ideals of $Q(i)$ of the form $(\frac{a+bi}{c})$ where $a^2+b^2=c^2$ with multiplication of ideals form a group.

Answer 1

These are the norm $1$ elements in $\Bbb Q(i)^*$. Call this group $G$. By Hilbert 90, each element of $G$ is $z/\overline z$ for some $z\in \Bbb Q(i)^*$. Thus there is a surjective homomorphism $\pi:\Bbb Q(i)^*\to G$ with $\pi(z)=z/\overline z$. Its kernel is $\Bbb Q^*$ and so $G \cong \Bbb Q(i)^*/\Bbb Q^*$.