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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

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    @QiaochuYuan Thanks for clarifying.2012-02-10

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$k(x)[y]$ 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 $uf+vg\in k[x]$.

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    @QiaochuYuan Ok I get it: Write $k[x,y]$ as $k[x][y]$. Then we already know that if two polynomials are coprime in a $A[y]$, $A$ a PID then they are coprime in $\operatorname{Frac}(A)[y]$. The result follows by setting $A = k[x]$.2012-02-10