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$\def\F{\mathbb F}$ $\def\Z{\mathbb Z}$ One reason that Chevalley-Warning theorem is that amazingly useful is the fact that for a finite field $\F$, any function from $\F^n$ to $\F$ is a polynomial. ...

Let's $P,Q\in\mathbb K[x]$, with $\mathbb K$ a finite field. On what necessary and sufficient condition on $R \in \mathbb K[x]$ is it : $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))...