TITLE: Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\setminus\{0\}$?
QUESTION [7 upvotes]: Let $k$ be a field. Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\setminus\{0\}$?

I can do it for specific polynomials, but I'm struggling to structure a coherent proof. Any hints would be greatly appreciated!
Thanks

REPLY [9 votes]: $k(x)[y]$ is a Euclidean domain, hence if $f,g$ are coprime in $k[x,y]$ they are coprime in $k(x)[y]$ and there are rational functions $U,V\in k(x)[y]$ such that $Uf+Vg=1$. Now multiply the denominators to get $uf+vg\in k[x]$.