Question: Let $k$ be any field, and $f; g \in k[x; y]$ irreducible polynomials, not multiples of one another. Write $K = k(x)$; prove that $f; g$ have no common factors in the PID $K[y]$.
My interpretation: I assume by no common factors, it is meant that $f,g$ are co-prime in $K[y]$ and thus we can find $a,b\in K[y]$ such that $af+bg=1$.
My thoughts: I would try to deduce it from definitions but I'm not sure how to manipulate the information that $f,g$ are irreducible polynomials of two variables because I cannot just say consider some field extension where they both factor linearly.
My only seemingly legitimate attempt would be: Let $h$ be the gcd of $f,g$ in $K[y]$ then there exist $a,b \in K[y]$ such that $af+bg=h$. I'm not sure where to go from here since I am not sure what I know about $f,g$ in $K[y]$. Are they irreducible still, etc...?
Extra Questions: Is $K[y]$ which is $k(x)[y]$ an extension of $k[x,y]$. How should I think about $k(x)[y]?$