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Given a polynomial $f \in \mathbb{C}[x_1,\ldots,x_n]$, then how can I prove that $f(a_1,\ldots,a_n) = 0$ implies $f$ is a summation of factors of $x_i - a_i$ for $i \in \{1,\ldots, n\}?$

This is not at all obvious to me. Any help would be much appreciated!

user26857
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user93826
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    It's not hard to show that $(x_1-a_1,\ldots,x_n-a_n)$ is maximal in $\mathbb{C}[x_1,\ldots,x_n]$. But, the ideal of functions vanishing at $(a_1,\ldots,a_n)$ is a proper ideal containing $(x_1-a_1,\ldots,x_n-a_n)$ and thus the two ideals must actually be equal. – Alex Youcis Oct 10 '13 at 17:50
  • I don't remember exactly but, this question is already asked and it is answered by Mr.Alex Youcis... –  Oct 10 '13 at 18:53
  • http://math.stackexchange.com/questions/500153/proving-that-kernels-of-evaluation-maps-are-generated-by-the-x-i-a-i/500167#comment1074479_500167 this might be helpful to you... –  Oct 10 '13 at 19:10

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