Let $f,g \in \mathbb{Q}[x,y]$ be coprime. I want to show that there are only finitely many common roots, i.e. only finitely many pairs $(a,b) \in \mathbb{C}^2$ with $f(a,b)=g(a,b)=0$. As a hint I should consider the gcd of $f,g$ over $\mathbb{Q}(x)[y]$.
If I could show that $gcd(f,g)=1$ considering $f, g$ in $\mathbb{Q}(x)[y]$ I can say that $res_y(f,g) \neq 0$. As $res_y(f,g) \in \mathbb{Q}(x)$ I have finitely many possibilities for the second component to get a root of $f$ and $g$. In the same way I can say that there are only finitely many possibilities for the first component by considering $\mathbb{Q}(y)[x]$ and the statement follows as all the common roots considered in $\mathbb{Q}[x,y]$ also have to be common roots in $\mathbb{Q}(x)[y]$.
My question is how I could show that $gcd(f,g)=1$ if I consider $f$ and $g$ in $\mathbb{Q}(x)[y]$? If I assume $gcd(f,g)=d \in \mathbb{Q}[x,y]$ (which I can as all non-zero polynomials $p \in \mathbb{Q}[x]$ are units in $\mathbb{Q}(x)[y]$), I get by definition $d \: | \: f, g$ in $\mathbb{Q}(x)[y]$. However I do not know how to get $d \: | \: f, g$ in $\mathbb{Q}[x,y]$.