Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$?

I know there are maximal ideals like $\langle X,Y-1\rangle$, $\langle X,Y\rangle$ containing $\langle X^{2}+Y^{2}-1\rangle$ but these are not proper differential ideals. If there exists a proper differential ideal containing $\langle X^{2}+Y^{2}-1\rangle$ then it must be generated by at least two polynomials say $f$, $g$. If any one the generator is of degree 1 or 2 then it is easy to see the ideal is not proper, but how to see for higher degrees of $f$ and $g$.


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    $\begingroup$ Welcome to MO. It is nice that you include a link to math stackexchange. However, if you feel that you might get an answer there (as you might for this question), then it is more common to wait a few days before asking the same question here. $\endgroup$ – Sebastian Goette Apr 20 '16 at 11:17

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