I want to know whether $Q[x,\sqrt{1-x^2}]$ is PID or not.
By using ring homomorphism theorem, I just need to know whether $Q[x,y]/x^2+y^2-1$ is PID or not.
I guess $x,y/x^2+y^2-1$ is not PID but I cannot formally prove this.
Thank you in advance!
And I heard a difficult answer.
'$Q[x,y]/x^2+y^2-1$ is Dedekind domain and it's Picard group is $\mathbb Z/2\mathbb Z$, so $Q[x,y]/x^2+y^2-1$ is not PID'.
If there is a more detailed explanation of this answer, I would appreciate it.
Consider the ideal $(X,Y)$ in the quotient. Intuitively, we can expect that this ideal is not principal because we only have the relation $X^2+Y^2=1$ and this does not allow us to eliminate one of the variables $X,Y$ in this ideal as generator. Now, try to formally prove this!