Maximal Ideals $\mathfrak m$ such that $\mathbb Q[x, y]/ \mathfrak m \cong \mathbb Q$

Question

I am asked to determine all maximal ideals $\mathfrak m \subset \mathbb Q[x, y]$ such that $\mathbb Q[x, y]/ \mathfrak m \cong \mathbb Q.$ I know that, for example, $(x, y)$ is a maximal ideal that fulfills this requirement. But how do I find every maximal ideal such that the statement holds?

Answer 1

Here's a hint to get you started.

By the first isomorphism theorem, an equivalent question is to find all the kernels of homormorphisms $\phi\colon\mathbb Q[x,y]\to\mathbb Q$.

But any such homomorphism is determined by the images of $x$ and $y$, since $$\phi(\sum a_{ij}x^iy^j) = \sum a_{ij}\phi(x)^i\phi(y)^j.$$

So suppose that $\phi(x) = a$ and $\phi(y) = b$.

Can you compute the kernel of $\phi$?