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Let $p$ be a univariate polynomial over a field $F$, and let $K$ be an extension of $F$.

If $p(x) = 0$ for all $x \in F$, does this imply that $p(x) = 0$ for all $x \in K$? How about if $p$ is multivariate?

For context, I'm trying to understand if doing Schwartz-Zippel-style arithmetic circuit identity-testing over a large enough extension field gives the right answer when the degree of the expression may be high.

user94741
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    For a finite field this test is never sufficient in general. It is sufficient if $|F|>\deg p$. Otherwise polynomial multiples of $x^{|F|}-x$ give you a headache. – Jyrki Lahtonen Oct 21 '14 at 21:08

2 Answers2

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No. Take $p(X)=X(X+1)$, $F=\Bbb F_2$, $K=\Bbb F_4$.

Indeed, let $\beta\in\Bbb F_4-\Bbb F_2$. We know that $\beta^2+\beta=1$, that is, $p(\beta)=1$. And $p(0)=p(1)=0$.

ajotatxe
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All you know is this: a nonzero polynomial of degree $d$ cannot have more than $d$ roots in some (extension) field.

orangeskid
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