# Polynomials that vanish on an arbitrary set of elements of a field

Can someone give me a hint how to solve the following (rather vague) exercise?

Let $S$ be a subset of $F^n$, where $F$ is a field and $F^n$ is the $n$-dimensional linear space over itself. Describe the set $L=\{p\in F[X_1,\ldots,X_n] \mid p \text{ vanishes on } S\}$ and give a geometric interpretation of $F[X_1,\ldots,X_n] /L$.

I looked in the Wikipedia page on formal polynomials in the hope to find something useful and it seemed to me that the Hilbert Nullstellensatz might be what is hiding behind this question, but couldn't adapt it, or even find a clear analogy (probably because I have absolutely no knowledge of the Nullstellensatz).

(Please bare also in mind, that my knowledge of algebra is limited to the basic definition of rings, ideals etc. plus some rather easy theorem about them - so nothing very deep.)

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Note that $p(x)\equiv q(x)\pmod{L}$ if and only if $p(x)-q(x)\in L$, if and only if for any $\mathbf{x}\in S$, $p(\mathbf{s})=q(\mathbf{s})$. – Arturo Magidin Nov 8 '11 at 18:22
@ArturoMagidin Ok, but how does that help me to describe $L$ ? – arila Nov 8 '11 at 18:35
Ok, I read about the zariski topology on wikipedia and it seems that there are some theorems that don't seem to hard to find a proof for that also give the basic properties of $L$, thus describing it. But could someone give me an idea for the geometric interpretation of ? – arila Nov 8 '11 at 18:36
@arila: It helps you give a geometric interpretation to $F[X_1,\ldots,X_n]/L$ in terms of functions that are evaluated at points of $S$. – Arturo Magidin Nov 8 '11 at 19:57

You could say that $L\subset F[X_1,\ldots,X_n]$ is a reduced ideal, reduced meaning that if $p^r\in L$ for some positive integer $r$, then $p\in L$. Can you see why this is true?
Now call $regular$ any function $S\to F$ given by a polynomial $p\in F[X_1,\ldots,X_n]$, that is a function of the form $s\mapsto f(s_1,...,s_n)$.
Try to prove that your mysterious quotient $F[X_1,\ldots,X_n] /L$ is isomorphic to the ring of regular functions on $S$.
Don't worry about the Nullstellensatz: it can't be applied anyway because $F$ is not assumed algebraically closed.