Prove that $6$ and $2(1+\sqrt{-5})$ do not have a gcd in $\mathbb{Z}[\sqrt{-5}]$

Question

Prove that $6$ and $2(1+\sqrt{-5})$ do not have a gcd. here those elements belong to $\mathbb{Z}[\sqrt{-5}]$.

They have common divisors like the number $2$ But if $a$ is another divisor it must divide the hypothetical gcd. What can i do?

Answer 1

Hint: $\mathbb{Z}[\sqrt{-5}]$ is not a UFD (Unique Factorization Domain), both $2,3,1+\sqrt{-5},1-\sqrt{-5}$ are irreducible elements in $\mathbb{Z}[\sqrt{-5}]$.

Answer 2

I found the answer 6 has as divisors $2,3, 1+\sqrt{-5}$ and $1-\sqrt{-5}$. Similarly, $2(1+\sqrt{-5})$ has as divisors $2,1+\sqrt{-5}$ but 2 does not divide $1+\sqrt{-5}$ nor the opposite so for the common divisors there is not one that is multiple of the others