Units in $\mathbb{Z}[\sqrt{d}]$

Question

I'm proving that if $d \in \mathbb{Z}, d < -1$ and $d$ is square-free , then the only units of $\mathbb{Z}[\sqrt{d}]$ are $\pm1$.

I proved it, but I never used that $d$ is square-free. Where should this come up?

Thanks!

Answer 1

The square-free part need not come up in a proof, since it is not required. You probably used the Brahmagupta Identity to show that the norm of a product is the product of the norms. That identity holds whether or not $d$ is square-free, and it is all that is needed for the proof.