Are the rings $\mathbb{Q}[x]$ and $\mathbb{Q}[x,y]$ principal ideal domains? Are the rings $\mathbb{Q}[x]$ and $\mathbb{Q}[x,y]$ principal ideal domains?
I understand what an integral domain is. I know the definitions of ideal and principal but have not ever dealt with principal ideal domains. I know that from the discussion here, $\mathbb{Z}[x]$ is not a principal ideal domain, but I'm not sure if the example with the ideal $(2,x)$ should extend to the polynomials with rational coefficients and would like some help.

$\mathbb{Q}[X]$ is a principal since $\mathbb{Q}$ is a field and you can divide. An ideal is generated by an element of lowest degree. But $\mathbb{Q}[X,Y]$ is not principal. The ideal generated by $(X,Y)$ does not have a single generator.
