Find all prime ideals in $K[X_1,X_2,X_3]$ which contain $(X_1^2-X_2X_3,\ X_1(1-X_3))$

Question

I have to determine $$\sqrt{(X_1^2-X_2X_3,\ X_1(1-X_3))}$$ in $K[X_1,X_2,X_3]$, where $K$ is a field.

I'm supposed to use the fact that $\sqrt{I}=\bigcap_{P\in\text{Spec}A,\ I\subseteq P}P$ for any ideal $I$ of a ring $A$.

My guess is that $(X_1^2-X_2X_3,\ X_1(1-X_3))$ itself is a prime ideal (correct me if I'm wrong please). To show this I wanted to find a morphism $\varphi:K[X_1,X_2,X_3]\rightarrow K[X] $ such that $\ker\varphi=(X_1^2-X_2X_3,\ X_1(1-X_3))$. I defined $\varphi$ via $\varphi(X_1)=X,\ \varphi(X_2)=X^2,\ \varphi(X_3)=1 $. But I am not sure if this $\varphi$ works. I'm having trouble finding its kernel.

Am I on the right track? Can someone give me a hint?

Thanks in advance!

Answer 1

Two (similar) possibilities:

1. Show that $I=(X_2,X_1)\cap (X_3,X_1) \cap (X_1^2-X_2,X_3-1)$ and then take the radical (Taking radicals commute with intersections).

2. Assume $\mathfrak{p}$ is a (minimal) prime containing $I$. Show that $\mathfrak{p}$ must be one of the three ideals above. For example, since $X_1(1-X_3)\in\mathfrak{p}$ you have $X_1$ or $1-X_3$ in $\mathfrak{p}$. Suppose $X_1\in \mathfrak{p}$. We also have $X_1^2-X_2X_3\in \mathfrak{p}$ and since $X_1$ is in there you also have $X_2X_3\in \mathfrak{p}$, so either $X_2$ or $X_3$ is in $\mathfrak{p}$ in particular this shows that in the case $X_1\in\mathfrak{p}$ we must have $\mathfrak{p}$ be either $(X_1,X_2),(X_1,X_3), (X_1,X_2,X_3)$ (But we can disregard the last one since it does not add anything to the intersection). Now do the same when $X_3-1\in\mathfrak{p}$.