I guess $X＝\mathbb Q[x,y]/(x^2+y^2-1)$ is not a principal ideal domain because we cannot eliminate two variables using $x^2+y^2＝1$. How can I formally prove this ?
My strategy of proof is to prove $(x,y-1/(x^2+y^2-1$ is not a principal ideal by contradiction.
If $(x,y-1)/(x^2+y^2-1)$ is a principal ideal generated by $f(x,y)$, then $●＝f(x,y)g(x,y)+(x^2+y^2-1)$, $f(x,y),g(x,y)∈\mathbb Q[x,y]$.
I attempt to find good $ ● ∈（x,y-1）/（x^2+y^2-1）$ and find contradiction.
Are there any good ideas?
Thank you very much in advance.
I think this is overkill but
$\Bbb{Q}[x,y]/(x^2+y^2-1)\cong \Bbb{Q}[\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}]$ and the latter is the subring of $\Bbb{Q}(t)$ of functions with no poles away from $\pm i$. It remains to check that no such function has only one zero at $t=0$ (because this function would have only one pole at $i$ or $-i$ which is possible in $\Bbb{Q}(i)(t)$ but not in $\Bbb{Q}(t)$)
Thus $(x,y-1)$ is not principal