I came across an argument on the lecture notes. It does not have a proof of the fact and I want to understand the argument.
Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$. Define $Z_f := \{(a,b) \in \bar{k} \times \bar{k} | f(a,b)=0\}$, $C_f :=\bar{k}[x,y]/(f)$ and $\bar{k}(Z_f)=Frac(C_f)$.
Then, $\bar{k}(Z_f)/\bar{k}(x)$ is a field extension of degree $f$.
I could not do anything much. I have few ideas like writing $f$ as $f(x,y)=a_n(x)y^n+\dots+a_1(x)y+a_0(x) \in \bar{k}[x][y]$. I do not know how to write the elements of $Frac(C_f)$ in a "nice way".
I mean better than writing this :$Frac(C_f)=\cfrac{p(x,y)+(f)}{q(x,y)+(f)}$.
So, I appreciate if you could enlighten me. How is that a field extension, what are the elements of it and what is the degree of $f$? I think the degree is the degree with respect to $y$ but not sure whether it is total degree or not.
Thank you.