When a two-generated ideal in $\mathbb{Z}[\sqrt{-3}]$ is the unit ideal or principal?

Question

I am trying to figure out when the ideal $(a_1+b_1\sqrt{-3},a_2+b_2\sqrt{-3}) $ is the unit ideal or principal in $\mathbb{Z}[\sqrt{-3}]$. Any hints?

Answer 1

We can do so directly as follows: Let $x+y\sqrt{-3} \in \mathbb{Z}[\sqrt{-3}]$. Multiply $a_1+b_1\sqrt{-3}$ by this number and set it equal to $a_2+b_2\sqrt{-3}$, then solve this system for $x$ and $y$ in terms of $a_1$ and $b_1$. When can this system be solved and what happens when it can (principal ideal case)? When can this system not be solved (unit case)?