Intersections of roots of polynomial in a field I tried to prove some property of fields, but I could not, and I hope someone can help me with that.  I have a question about fields and roots.
If I have an arbitrary family (each of them is a set of roots $(x,y)$ of a polynomial in two variables over an algebraically closed field), is the intersection of all of them the set of roots of another polynomial?
The set $X$ is an algebraically closed set, the set of roots in this 2 variables polynomial is clearly of the form $(x,y)$.  
Do I need to know some special property of algebra to prove this, or only the definition of algebraically closed set?
 A: I mentioned in an earlier, less relevant answer that the intersection $\{(0, 0)\}$ of the zero sets of $x$ and $y$ was not the zero set of a single polynomial. One way to see this is via the following lemma.

Let $k$ be an algebraically closed field, and let $f \in k[x, y]$ be a non-constant polynomial. Then the zero set of $f$ is infinite.

Here are some hints toward proving this. Since $f$ is not a constant, some variable — which might as well be $x$ — appears in it. Then we can write
$$
f(x, y) = \sum_{i = 0}^n g_i(y)x^i
$$
where each $g_i \in k[y]$, and $n > 0$ is such that $g_n$ is not the zero polynomial. The key is that $g_n$ has only finitely many zeros in $k$. Can you take it from there?
A: Your set X can be given as the roots of a finite set of polynomials, but possibly not a single one (for example if X is the empty set). 
There's a bit of abstract algebra that deals specifically with this kind of question. The key point is that if $F$ is an algebraically closed field, then $F[x,y]$ is "Noetherian", which means that any ideal is finitely generated. 
What does this have to do with your question? Suppose you have two polynomials $f$ and $g$.
Notice that a point $(a,b)$ is a root of polynomials $f$ and $g$ if and only if it is a root of the polynomial $p(x,y) f(x,y) + q(x,y)g(x,y)$, for any polynomials $p$ and $q$. In other words, it needs to be a root of all the polynomials in the ideal generated by $f$ and $g$. 
Therefore instead of considering your family of polynomials, it's enough to consider the ideal generated by that family. Since $F[x,y]$ is Noetherian, that ideal is finitely generated, and those generators are the polynomials you're looking for. 
You should be able to read about this in any textbook on abstract algebra/ring theory.
