We prove the <polynomial nonvanishing below the field size> by <mathematical induction> on $n$. The case $n=1$ is the <Lagrange root bound over a field>: a nonzero polynomial of degree below $p$ cannot have all $p$ elements of $\mathbb F_p$ as roots.

For the induction step, suppose that $P$ vanishes on all of $\mathbb F_p^n$ and write
$$
P(x_1,\ldots,x_n)=\sum_{j=0}^{p-1}P_j(x_1,\ldots,x_{n-1})x_n^j.
$$
The upper limit is valid because the <total degree of a polynomial> is less than $p$. Fixing the first $n-1$ variables produces a univariate polynomial of degree below $p$ which vanishes at every $x_n\in\mathbb F_p$. It is therefore the zero polynomial, so every $P_j$ vanishes on all of $\mathbb F_p^{n-1}$. Each $P_j$ also has total degree below $p$, and the induction hypothesis gives $P_j=0$ for every $j$. Thus $P=0$.

Taking the <contrapositive>, every nonzero such $P$ has some $a\in\mathbb F_p^n$ for which
$$
\boxed{P(a)\ne0.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 147 / 1 / iii / Solution

Let $P\in\mathbb F_p[x_1,\ldots,x_n]$ have <total degree of a polynomial> less than $p$. If the associated <polynomial function> vanishes on all of $\mathbb F_p^n$, then $P$ is the zero polynomial. Induct on $n$: write
$$
P=\sum_{j=0}^{p-1}P_j(x_1,\ldots,x_{n-1})x_n^j.
$$
For each fixed $(x_1,\ldots,x_{n-1})$, the resulting univariate polynomial of degree less than $p$ has all $p$ field elements as <roots of a polynomial>[roots], so every coefficient $P_j$ vanishes everywhere. The induction hypothesis makes every $P_j$ the zero polynomial.

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