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

Question

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 $(x, y)$.

Answer 1

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?

Answer 2

The full ring cannot be Euclidean; on the other hand, there is a preservation of degree. The degree of a single term $x^a y^b$ is $a+b.$ The degree of a sum of terms is the highest degree term.

Then $$\deg (fg) = \deg f + \deg g$$

Let's see, a nonzero constant has degree $0.$ We don't define a degree of $0$ itself, or sometimes call that $-\infty$