For an algebraicaly closed field $k$ are all finite subsets of Affine $n$-space $A^{n}\left(k\right)$ algebraic sets (here for $n>1$), and if so, for a given finite set $X\subset A^{n}\left(k\right)$ what set of polynomials in $k\left[x_{1},\dots,x_{n}\right]$ has $X$ as its common zero set? This probably has an answer, I just don't know how to phrase the question so as to get an answer by internet search engine.

Also if I have a set of polynomials $P_{1},\dots,P_{k}\in\mathbb{Q}\left[x_{1},\dots,x_{n}\right]$ such that the common zero set $Z(P_{1},\dots,P_{k})$ is a finite set of size $m$, i.e. $X=\{(z_{11},\dots,z_{1n}),\dots,(z_{m1},\dots,z_{mn})\}$, does this imply that each coordinate $z_{ij}$ is algebraic over $\mathbb{Q}$. I asked someone this question, and he felt that each $z_{ij}$ should turn out to be algebraic, but I can't see how this is proved?

Algebraic geometry is not my area, I just was wondering about these questions and am having a hard time tracking down an answer on the internet.