Maximal Ideals of $\mathbb C[x, y]$

Question

I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$.

I am unable to prove it.

So I considered an easier version of the problem.

Let $M=(p, q)$ be an ideal in $\mathbb C[x, y]$, where $p$ and $q$ are elements of $\mathbb C[x, y]$. If $M$ is maximal, then $\deg p=\deg q=1$.

I am stuck even at this.

An ideal is maximal if quotienting the parent ring with it gives a field, thus $\mathbb C[x, y]/(p, q)$ is a field. But I am having no ideas.

Answer 1

You have to use Hilbert's Nullstellensatz that's is a polynomial say $p$ in $\mathbb C[x,y]$ have a zero that a point $(a,b)\subset \mathbb{C}^2$,where it vanish in this case it imply the ideal generated by the polynomial $(p)\subset (x-a,x-b)$. Now you can prove.......