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I can see how the power sum is applicable in the final paragraph of the proof, but it's still too vague in my head. Can anyone provide more details as to why the power sum of each monomial vanishes? The lemma in reference can be found here. When we have $1$ variable for a power sum, it is much more obvious to me, but when we have $K^n$ it is not.

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user240033
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  • What is the sum $\sum_{a \in \mathbb{F}_q},a^r$ for $r=0,1,2,\ldots,q-2$? – Batominovski Jul 10 '15 at 08:54
  • @Batominovski, it's $0$. That's what the lemma I linked proves, and that's what I mention is clear, in contrast to how the monomials have several variables instead and it still vanishes, i.e. that isn't clear. – user240033 Jul 10 '15 at 09:27
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    $\sum_{\left(a_1,a_2,\ldots,a_k\right)\in\mathbb{F}q^k},a_1^{r_1}a_2^{r_2} \cdots a_k^{r_k} = \left(\sum{a_1\in \mathbb{F}q} a_1^{r_1}\right) \left(\sum{a_2\in \mathbb{F}q} a_2^{r_2}\right)\cdots \left(\sum{a_k\in \mathbb{F}_q} a_k^{r_k}\right)$. – Batominovski Jul 10 '15 at 09:34
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    Wow, that was ridiculously simple. That answers my question perfectly. – user240033 Jul 10 '15 at 09:36

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