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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?

Charles Fu
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    Do you have any examples of nonidentical equivalent polynomials over a field of characteristic zero? or even over an infinite field of positive characteristic? – Gerry Myerson Nov 08 '13 at 23:18
  • No, even for univariate polynomials, in a field of finite characteristic it's possible for a degree 1 polynomial to be "equivalent" to a reducible polynomial. – hardmath Nov 08 '13 at 23:18

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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.

Gerry Myerson
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    Indeed I found the solution at http://math.stackexchange.com/questions/23241/identically-zero-multivariate-polynomial-function following your remark; thanks! – Charles Fu Nov 08 '13 at 23:39