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.)