Unique maximal ideal containing $(X,Y^3)$

Question

I want to show that there is a unique maximal ideal in $K[X,Y]$ which contains the ideal $(X,Y^3)$, namely $(X,Y)$. Can anyone give me a hint?

Answer 1

I assume that $K$ is a field in your question.

The ideals of $K[X,Y]$ containing $(X,Y^3)$ are in bijection with the ideals of $K[X,Y]/(X,Y^3)$, and this bijection preserves inclusions. In view of this, what you want to show is that the ring $K[X,Y]/(X,Y^3)$ contains a unique maximal ideal (such rings are called local rings).

To do this, one could first show that $K[X,Y]/(X,Y^3)$ is isomorphic to $K[Y]/(Y^3)$, and then prove that this last ring is local (its only maximal ideal is generated by $Y$).

Answer 2

Hint:

A maximal ideal is prime. On the other hand $K[X,Y]/(X,Y)\simeq K$.