Consider the finite field $F_q$, where $q$ is a power of an odd prime and $N$ is a power of $q$. We have a homogeneous symmetric polynomial $$ E_{l,s}(x) = \sum_{\substack{l_1+l_2+\cdots +l_s=l \\ l_i\geqslant 1}}\ \sum_{1\leqslant i_1<i_2<\cdots <i_s\leqslant N}x_{i_1}^{l_1}x_{i_2}^{l_2}\cdots x_{i_s}^{l_s} \, , $$ where $l=q^a-1$ ($a\in \mathbb{N}$) and $l-q+2\leqslant s \leqslant l$. We can write it in terms of elementary symmetric polynomials $e_1, e_2, \ldots, e_l$, and it is easy to check the coefficient of $e_l$ is a constant in $F_q$. The question is how to find out this constant.

I guess it is $1$ for all $l-q+2\leqslant s \leqslant l$, and I verified it for $s=l,l-1,l-2$. However, I cannot find a way to prove it.

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