Let $\mathbb {F}$ be a field.
Let $p(x),q(x) \in \mathbb{F}[x] s.t. \forall a \in \mathbb{F},p(a)=q(a) $.
Is it true, in general, that $p(x)|q(x) \vee q(x)|p(x)$?
It seems plausible for me that this holds even in finite fields, as I expect the result of the division to be 1, but I've tried some examples, dividing via Ruffini's method, and this seems not to hold, but I can't figure out why.
Thank you.
In a finite field $F$ of $q$ elements, the polynomial $f(x)=x^q-x$ has the property that $f(a)=0$ for all elements of $F$. Therefore so do $xf(x)$ and $(x+1)f(x)$.