Algebraic Geometry and Maximal ideals I am solving the following problem but couldn't figure out a strategy to solve:
Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ideal generated by the polynomial $y^2+x^3-17$? What is a common strategy to solve this kind of problems?
One thing that I noticed was the following: If we solve $x^3-17=0$ and $y^2=0$ then the solutions to these equation when plugged in the equation $y^2+x^3-17$ yields 0. I am not sure what it means. Does that mean that $I$ is indeed the whole ring $R=\mathbb{C}[x,y]$?

Notice that $(y^2+x^3-17)\subset (y^2,x^3-17)$. Therefore, the quotient of the ring $\mathbb{C}[x,y]/I$ by the ideal, generated by $(y^2,x^3-17)$ is actually isomorphic to $\mathbb{C}[x,y]/(y^2,x^3-17)$. All I am using here is that for any ring $A$ and ideals $I\subset J$, we have an isomorphism $(A/I)/(J/I)\simeq A/J$. So the question boils down to determining if $\mathbb{C}[x,y]/(y^2,x^3-17)$ is a field. It is obviously not. So the answer is "no".
