Describe the closure in the Zariski topology I have a little bit to fight with the Zariski - topology. In $\mathbb{C}^2$, i have to describe by a finite number of polynomial equations the Zariski closure of: 


*

*$C:= \{(n,n^2) \in \mathbb{C}^2 : n \in \mathbb{N} \}$

*$D:= \mathbb{C}^2 \setminus \{(0,0)\}  $.


We can use the fact that for $M \in \mathbb{C}^n: Z(I(M))= \bar{M}$, where $\bar{M}$ denotes the closure of $M$ in the Zariski topology.

  
*
  
*Denote $x = n$ and $y = n^2$. It follows
  $$
 x^2 - y = 0 \Rightarrow x^2 - y \in I(C) \Rightarrow \bar{C} = Z(I(C)) \subsetneq \mathbb{C}^2,
$$
  since $I(C) \neq {0}$. 
  

So now i don't know how to find a finite number of polynomial equation. I think, but i am not sure, that we can conclude that $I(C) = (x^2 -y)$ and thus $\bar{C} = Z((x^2-y))$. Am i right?


  
*Suppose $D\subseteq Z(J)$ for an ideal $J \subseteq \mathbb{C}[X,Y]$ and $f \in J$ with $f(a) = 0 \ \forall a \in D$.
  

I think we can't deduce that $f$ is the zero polynomial. But my intuition says that $\bar{D} = Z(I(D)) = \mathbb{C}^2$. Where is/are my mistake/s? 
Thanks in advance.
 A: Perhaps the issue is that you're not comfortable with the Nullstellensatz and dimension theory?
Here's how to do the first problem.
For 1 you've shown that $\overline{C} \subseteq V(y-x^2)$.  Since $k[x,y]/(y-x^2) \cong k[x,x^2] = k[x]$ is an integral domain, the ideal $(y-x^2)$ is prime, and hence the parabola $V(y-x^2)$ is irreducible.  Moreover, it is isomorphic to the affine line $\mathbb{A}^1$, and hence one-dimensional.  By the Nullstellensatz, then, any closed set properly contained in $V(y-x^2)$ is zero-dimensional, and hence a finite union of points.  Such a set can't contain all of $C$, so $V(y-x^2)$ is indeed the closure of $C$.
Of course, this is using a fairly sophisticated understanding of algebraic geometry to establish that the curve is actually one-dimensional.  Depending on how you're doing the class you may not have access to this machinery (namely, that dimension and affine-ness are intrinsic notions and that an affine variety may be recovered (up to isomorphism) from its coordinate ring), and consequently may have to establish the dimension of $V(y-x^2)$ in some other way.
