Question about some aspect of Zariski topology

Question

In the Folland book "real analysis", in the page 117 is written about Zariski topology: Let $k$ be a field and let $k(X_1,...,X_n)$ be the ring of polynomials in $n$ variables over $k$. Each $P$ from $k(X_1,...,X_n)$ determines a polynomial map $p:k^n \to k$ by substituting elements of $k$ for the formal indeterminates $X_1,...,X_n$. The correspondence $P \to p$ is one-to-one precisely when $k$ is infinite.

As I understand if $k$ is finite, then the correspondence is not one-to-one. I am not sure that for me that sentence is clear. Please help me.

Answer 1

Here is an example: if $k=\mathbb F_2$ (the field with two elements), then $x^2-x$ is a polynomial with degree $2$; in particular, it is not the null polynomial. But the polynomial function$$\begin{array}{ccc}\mathbb F_2&\longrightarrow&\mathbb F_2\\x&\mapsto&x^2-x\end{array}$$is the null function.

More generally, if $k$ is a finite field whose elements are $a_1,\ldots,a_n$, then $(x-a_1)\times\cdots\times(x-a_n)$ is not the null polynomial, but the polynomial function associated with it is the null function.

Answer 2

For example, if $k=\mathbb{F} _2$, the polynomials $X$ and $X^2$ determine the same function $k\to k$.