Exploring Algebraic Sets: Finding Irreducibility and Prime Ideals

[evinda]

Hello! (Smile)

Notice that in $\mathbb{C}[X,Y,Z]$:
$$V(Y-X^2,Z-X^3)=\{ (t,t^2,t^3)/ t \in \mathbb{C}\}$$

In addition, show that:

$$I(V(Y-X^2,Z-X^3))=<Y-X^2,-X^3>$$

Finally, prove that the ideal $<Y-X^2,Z-X^3>$ is a prime ideal of $\mathbb{C}[X,Y,Z]$. Conclude that the algebraic set $V(Y-X^2,Z-X^3)$ is irreducible.

Could you give me some hints to solve the above exercise? (Thinking)

 

[evinda]

Notice that in $\mathbb{C}[X,Y,Z]$:
$$V(Y-X^2,Z-X^3)=\{ (t,t^2,t^3)/ t \in \mathbb{C}\}$$

Could we do it maybe like that?

$$V(Y-X^2, Z-X^3)=\{(a,b,c) \in \mathbb{C}^3 | b-a^2=0, c-a^3=0 \Rightarrow b=a^2, c=a^3\}=\{(t, t^2, t^3)| t \in \mathbb{C}\}$$

In addition, show that:

$$I(V(Y-X^2,Z-X^3))=<Y-X^2,-X^3>$$

Is it like that?

$$I(V(Y-X^2, Z-X^3))=I(\{(t, t^2, t^3)|t \in \mathbb{C}\})=\{f(X,Y,Z) \in \mathbb{C}[X,Y,Z]|f(t,t^2,t^3)=0\}\overset{*}{=}\{(Y-X^2) \cdot g(X,Y,Z)+(Z-X^3) \cdot h(X,Y,Z) | g,h \in \mathbb{C}[X,Y,Z]\}=\langle Y-X^2, Z-X^3\rangle $$

At $(*)$ can we say it like that because we know from $V(Y-X^2, Z-X^3)=\{(t,t^2,t^3)|t \in \mathbb{C}\}$ that $(t,t^2,t^3)$ is a solution of $Y-X^2=0$ and $Z-X^3=0$ ?

(Thinking)