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Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S(x) \in \mathbb{E}[x]$. How I will be able to reduce a expression: $\sum_{i,j=0}^{r-1}p_{ij}(\sum_{k=0}^{n-1}s_kx^{q^k})^{q^i+q^j}$?

juaninf
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    Use the facts $a^{q^i+q^j}=a^{q^i}\cdot a^{q^j}$, and $$(\sum_k b_k)^{q^i}=\sum_k b_k^{q^i}.$$ You get $$ \sum_{i,j}p_{ij}(\sum_k s_k^{q^i}x^{q^{k+i}})(\sum_k s_k^{q^j}x^{q^{k+j}}). $$ – Jyrki Lahtonen Feb 05 '14 at 13:11
  • With your advice I get fix the associate question http://math.stackexchange.com/questions/650176/univariate-and-matrix-representation-of-affine-transformation/650500#650500. Very thanks, here the demonstration http://juaninf.blogspot.com.br/2014/02/tranformacao-afim-mpkc.html – juaninf Feb 06 '14 at 18:51

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