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
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....