Consider the polynomial ring $\mathbb{C}[X,Y]$ and the ideal $I$ generated by $Y^2-X$. How many maximal ideals are there in Quotient ring $\displaystyle\frac{\mathbb{C}[X,Y]}{I}$

solution i tried-Here the given $\mathbb{C[X,Y]}$ is a principal ideal domain ,and the elements of $\displaystyle\frac{\mathbb{C}[X,Y]}{I}$ will be of form $\sum_{i,j}a_kX^iY^j+\langle Y^2-X \rangle$ but the thing is that ,i don't know how to use this to find maximal ideals of this given quotient ring

please help

  • $\begingroup$ i edited the question,i got the point that C[X,Y] is not pid $\endgroup$ – TheStudent Feb 12 at 16:46
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    $\begingroup$ Do you know (one of the many forms of) the Nullstellensatz? I suppose you don't need its full strength for this problem, but it will tell you exactly what the maximal ideals of your quotient ring are... $\endgroup$ – Alex Wertheim Feb 12 at 17:49
  • $\begingroup$ can you please make it more simple,? $\endgroup$ – TheStudent Feb 13 at 10:55

The map $P(X,Y)\in\mathbb{C}[X,Y]\mapsto P(T^2,T)\in\mathbb{C}[T]$ is a surjective ring morphism with kernel $I$, so your quotient ring is isomorphic to $\mathbb{C}[T]$, which has infinitely many maximal ideals, namely the ideals $(T-a), a\in\mathbb{C}$.

I leave the details to you....

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