Consider the integral domain $\displaystyle \mathbb{Z}\[\sqrt{-13}\] = \{ x + y \sqrt{-13} | x,y \in \mathbb{Z} \}$.

Show that $\displaystyle 2$ and $\displaystyle 3 + \sqrt{-13}$ have no common factors in $\displaystyle \mathbb{Z}\[\sqrt{-13}\]$ except for $\displaystyle 1$ and $\displaystyle -1$.

I can show that both $\displaystyle 2$ and $\displaystyle 3 + \sqrt{-13}$ are irreducible. However, is there another way to prove that $\displaystyle 1$ and $\displaystyle -1$ are the only common factors?