Let $a$ be an element of order $n$ in $GF(2^q)$. Let $b$ be a non-zero element of $GF(2^q)$ such that $b\not \in \{r,r^2,\cdots ,r^n\}$. Assume that $x_i=(1+ba^i)$ for $1\leq i \leq n$.
Now consider the elementary symmetric polynomial of degree $n-1$ which is defined by $$ e_{n-1}(x_1,x_2,\cdots,x_{n})=\sum_{1\leq j_1<j_2<\cdots <j_{n-1} \leq n}x_{j_1}x_{j_2}\cdots x_{j_{n-1}} $$
Question: How to show that $e_{n-1}(x_1,x_2,\cdots,x_{n})=1$?
I can prove that for $1\leq k \leq n-1$ we have $p_k(x_1,x_2,\cdots,x_{n})=\sum_{i=1}^nx_i^k=1$. But I don't know how to continue it.
Thanks for any help
$n$ is odd and $$\prod_{i=1}^n (Y-1-b a^i) = (Y-1)^n -b^n= \sum_{l=0}^n {n \choose l} Y^l (-1)^{n- l} - b^n$$