If a polynomial is zero on a field F, is it zero on every extension of F?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 ∈ F, does this imply that p ( x ) = 0 for all x ∈ 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.

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If a polynomial is zero on a field F, is it zero on every extension of F?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  ∈  F, does this imply that   p  (  x  )  =  0 for all   x  ∈  K? How about if p is multivariate?For context, I&#039;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.

embutiridsl · Accepted Answer

Step 1No. Take   p  (  X  )  =  X  (  X  +  1  ),   F  =            F        2  ,   K  =            F        4  Step 2Indeed, let   β  ∈            F        4    −            F        2  . We know that       β    2    +  β  =  1, that is,   p  (  β  )  =  1. And   p  (  0  )  =  p  (  1  )  =  0.

If a polynomial is zero on a field F, is it zero on every extension of F? Let p be a univariate polynomial over a field F, and let K be an extension of F.

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