Show that it is an integral domain

Question

I want to show that

$$\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3) \text{ is an integral domain }.$$

How can I do this? Do I have to find a homomorphism from $\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3)$ to an integral domain?

Is there also another way?

Answer 1

Define a $\mathbb{C}$-algebra homomorphism $F$ from $\mathbb{C}[X,Y,Z]$ to $\mathbb{C}[X]$ by $F(X) = X$, $F(Y) = X^2$, and $F(Z) = X^3$.

Let $I = (Y-X^2,Z-X^3)$. It is clear that $Y-X^2, Z-X^3 \in \ker F$, hence $I \subseteq \ker F$.

On the other hand let $g(X,Y,Z)$ be an arbitrary polynomial in $\mathbb{C}[X,Y,Z]$. It is clear that $g$ is congruent modulo $I$ to some polynomial $h(X)$. Therefore $g \in \ker F$ if and only if $h \in \ker F$. But $h \in \ker F$ if and only if $h = 0$. Thus whenever $g \in \ker F$, we have $g \in I$. This proves $\ker F = I$.

It follows that $F$ induces an isomorphism between $\mathbb{C}[X,Y,Z]/I$ and its image under $F$, which is $\mathbb{C}[X]$.