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?