Is it possible that the zeroes of a polynomial form an infinite field?

Question

Let $K/F$ be a finite field extension and suppose that $F$ is infinite. Is it possible to have a nonzero polynomial $p \in K[x_1,...,x_n]$ that vanishes in $F^n$?

Answer 1

Proof by induction on $n$: For $n=1$, we could divide off a linear factor $x_1-a$ for each $a\in F$, which is impossible. For each $a\in F$, the polynomial $p(x_1,\ldots,x_{n-1},a)\in K[x_1,\ldots,x_{n-1}]$ vanishes in $F$, hence by induction we may assume that it is the zero polynomial. When we write $p$ as element of $K[x_n][x_1,\ldots,x_{n-1}]$, this means that the coefficients of each monomial is a polynomial $\in K[x_1]$ that vanishes on $F$. As in the case $n=1$, this implies that all coefficients are zero.