# Polynomial Ring $\mathbb{C}[X,Y]$

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

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 $$(T-a), a\in\mathbb{C}$$.