> 
> Prove the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$ .
> 
> 
> 

This argument [Prove that the class number of $\mathbb{Z}[\zeta\_3]$ is $1$](https://math.stackexchange.com/questions/23086/prove-that-the-class-number-of-mathbbz-zeta-3-is-1) doesn't work since $\mathbb{Z}[\sqrt{-5}]$ is not a unique factorization domain. How do we prove this?