Question regarding g.c.d of two polynomials

Question

Let $f,g \in C[x,y]$ be two non-constant polynomials with no common factor. I want to prove that in the Euclidean domain $C(x)[y]$ the g.c.d of $f$ and $g$ is in $C(x)$ and it looks like I'm missing something easy while proving it. Thanks in advance!

Answer 1

There's nothing special about $\mathbb{C}(x)$. If $f$ and $g$ are polynomials in the Euclidean domain $\mathbb{K}[y]$ and their gcd is not in $\mathbb{K}$ then the gcd is a non-trivial common factor. That holds just as well in the particular case that $\mathbb{K}$ is itself a polynomial ring.