This feels like a trivial question but somehow I couldn't come up with an immediate solution. Given polynomial $P$ in a multivariate polynomial ring over some base field $F$, if $P$ is irreducible over $F$, does it follow that any equivalent polynomial is also irreducible over $F$? If not, is there an explicit counterexample? By equivalent we mean taking the same value as $P$ on all possible variable assignments.
Just realized this probably won't work for finite field (e.g. $(x^2 + 1)^2 \equiv 1$ in $F_3$), but what about characteristic 0 fields?
If two polynomials are equivalent, then their difference is identically zero. There is no such univariate polynomial over an infinite field, except the zero polynomial. By induction on the number of variables, I expect you can prove a similar result in general.