Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element. 
Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element.

I know the ideal $(x,y)$ is maximal since $\mathbb Q[x,y]/(x,y) \cong \mathbb Q$; and I know $\mathbb Q[x,y]$ is not a P.I.D.
But I am not sure how to explicitly explain why there is not single generator for the ideal $(x, y)$.

If $(x,y) = (p(x,y))$ then it follows $p|x$ and $p|y$ which is an easy contradiction, or am I completely misunderstanding something?
