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.
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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 $G \cong \Bbb Q(i)^*/\Bbb Q^*$.
Angina Seng
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