We have the integral domain, for $k \ge 2, k \in \mathbb{Z},$ $$\mathbb{Z}[\sqrt{k} i] = \{m + n \sqrt{k} i : m,n \in \mathbb{Z}\}$$ and a Euclidean function defined by $$v:\mathbb{Z}[\sqrt{k} i] \to \mathbb{N}_0;\;\; v(m + n \sqrt{k} i) = m^2 + kn^2$$ for $m, n \in \mathbb{Z}$. Now let $a_1, \dots ,a_n \in \mathbb{Z}[\sqrt{k} i]$ for some $n \in \mathbb{N}$ and suppose $v(a_1), \dots ,v(a_n)$ are coprime in $\mathbb{Z}$. Show that $a_1, \dots ,a_n$ are coprime in $\mathbb{Z}[\sqrt{k} i]$.
I'm not sure how to proceed with this question. I can write $v(a_1), \dots ,v(a_n)$ as a linear combination of 1, but I don't know how to show the $a_1, \dots ,a_n$ as a similar linear combination.
Please help!
Hint $\,\ v(\alpha) = \alpha\bar \alpha,\ $ so $\,\ 1 = \gcd(v(\alpha), v(\beta)) = \gcd(\alpha\bar \alpha, \beta\bar \beta)\ \Rightarrow\ 1 = \gcd(\alpha,\beta) $