Could you tell me if the following step is correct??
If $p(x, y, z) \in ker \phi$, $$\phi(p(x, y, z))=0 \Rightarrow \phi(g(x, y, z)(y-x^2)+(z-x^3)a(x, z)+b(x))=0 \Rightarrow \phi(g(x, y, z)) \phi((y-x^2))+\phi((z-x^3))\phi(a(x, z))+\phi(b(x))=0\Rightarrow \phi(b(x))=0 \Rightarrow b(\phi(x))=0 \Rightarrow b(x)=0$$
In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz??
EDIT:
To show that $\langle y-x^2, z-x^3\rangle$ is prime do we have to do the following??
We co...
If $V$ is an algebraic set of $K^n$, show that $V$ is irreducible exactly then when
$I(V )$ is a prime ideal of $K[X_1, X_2, \dots, X_n]$ .
Let $V$ be irreducible.
We suppose that $I(V)$ is not a prime ideal of $K[X_1, X_2, \dots, X_n]$.
That means that there are polynomials $f,g \in K[X_1, ...
In my lecture notes there is the following example for intersection points of curves:
$$F(x, y, z)=xz^3-y^4 \\ G(x, y, z)=xz^2-y^3$$ in $\mathbb{P}^2(\mathbb{C})$, where $\mathbb{P}^2(\mathbb{C})=U_2 \cup H$
where $U_2=\{[x, y, 1] | x, y \in \mathbb{C}\}, H=V(z)=\{[x, y, z] | x, y \in \mathbb...
