Let $k$ be an algebraically closed field. I have proven that all maximal ideals of $k[x,y]$ are of the form $m=(f,g)$, where $f\in k[x]$ monic and irreducible and $g\equiv h$ mod $f$ with $h$ irreducible in $k_{f}[y]=\left(k[x]/(f)\right)[y]$. Now I want to use this result to find a bijection $\varphi:k^2\tilde{\to}$SpecMax($k[x,y])$.
I was thinking of $k^2\ni(\alpha,\beta)\mapsto (f^{\alpha}_{k[x]},f^{\beta}_{k[x,y]}\text{ mod }f^{\alpha}_{k[x]})\subset k[x,y]$ as maximal ideal, where $f^{\alpha}_{k[x]}$ denotes the unique minimal polynomial of $\alpha$ over $k[x]$. Is this going in the right direction?
Well, the maximal ideals are of the form $\langle x-a,y-b\rangle$ for $a,b\in k$ and so the isomorphism is quite obvious.