Exercise 10.1

Proof. Assume that $f$ vanishes on all of $A_n(K)$. We have to prove that $f$ is the zero polynomial.

The proof is by induction on $n$. If $n=1$, then $f$ is a polynomial with one variable, which vanishes on $A_1(K)=K$. Since $K$ is infinite, $f$ have more than $d$ roots, where $d=\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(f)$, thus $f$ is the zero polynomial.

Suppose that we have proved the result for $n-1$ and write

where the $x_i$ are variables, and $g_i$ are polynomials in $x_1,\dots ,x_{n-1}$.

For all $(a_1,\dots ,a_n)\in K^n$,

From the result for $n=1$, we obtain that the polynomial ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^{s-1}{g}_i(a_1,\dots ,a_{n-1})x_n^i$ is null, thus for all $(a_1,\dots ,a_{n-1})\in K^{n-1}$,

The induction hypothesis shows that $g_i(x_1,\dots ,x_{n-1})=0$, thus $f(x_1,\dots ,x_n)=0$. □