How do I use the Euclidean Algorithm in the ring $\mathbb Z[\sqrt{3}]$? I was asked to find the GCD of $7+\sqrt{3}$ and $6-2\sqrt{3}$ in the ring $\mathbb Z[\sqrt{3}]$, but have no idea where to start. Any tips would be appreciated!

I didn't use the Euclidean algorithm but here's my approach: 
$6-2\sqrt{3}=2(3-\sqrt{3})=2\sqrt{3}(\sqrt{3}-1)$,  $ 2=-(1-\sqrt{3})(1+\sqrt{3})$ 
Therefore: $$6-2\sqrt{3}=(1-\sqrt{3})(1+\sqrt{3})\sqrt{3}(1-\sqrt{3})$$
On the other hand: $$7+\sqrt{3}=7+3\sqrt{3}-2\sqrt{3}=(1+\sqrt{3})(1+2\sqrt{3})+(1-\sqrt{3})(1+\sqrt{3})\sqrt{3}=(1+\sqrt{3})(-2+3\sqrt{3})$$
You can check that the elemnts in both factorizations are irreducible and since our integral domain is a principal ideal domain, the elements are prime as well. So the GCD is $1+\sqrt{3}$. Can anyone confirm my result?
