Precisely which polynomials vanish on $\mathbb{F}_{p^n}$?

Question

Let $F$ be a field. Let $V \subseteq F^k$. Let $\mathcal{I} (V)$ be the ideal in $F[t_1, \ldots, t_k]$ of polynomials that vanish on $V$.

If $F=\mathbb{F}_{p^n}$ is a finite field, what is $\mathcal{I}(F)$? (That is, I am asking about $k=1$ and $V= F$. If anyone knows a characterisation of $\mathcal{I}(F^k)$ for larger $k$ I'd be delighted to see it.)

We know at least that $x^{p^n}-x \in \mathcal{I}(F)$, that is, $x^{p^n}-x$ vanishes on $F$. Is $\mathcal{I}(F)$ generated just by $x^{p^n}-x$, or are there other elements?

Answer 1

Indeed $\mathcal{I}(F)=(x^{p^n}-x)$ because if $f\in F[t]$ is such that $f(c)=0$ for all $c\in F$, then $x-c\mid f$ for all $c\in F$ and so $\prod_{c\in F}(x-c)\mid f$, where $\prod_{c\in F}(x-c)=x^{p^n}-x$.