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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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    @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 $S$.2011-11-08

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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.
And also: there are zillions of explicit, elementary and well-formulated exercises for beginners in algebraic geometry. I 'm not sure that this is one of them. So don't be discouraged if you couldn't solve it!