Theorem 1 (Chevalley–Warning)

Let K be a finite commutative field of characteristic $p$. Let $n$ be an integer $\geqslant 1$ and $(f_i)_{i\in I}$ a finite family of nonzero elements of $K[X_1, . . . ,X_n]$. Denote by Z the set of elements $\mathbf{x}$ of $K^n$ such that we have $f_i(\mathbf{x}) = 0$ for $i\in I$. If we have $n >\sum_{i\in I}$ deg($f_i$), then the cardinal of Z is divisible by $p$.