Obtaining some elements of Galois field $2^q$ Consider an $n$-tuple $(\alpha_1,\alpha_2,\cdots, \alpha_n)$ where $\alpha_i$, $1\leq i \leq n$, are elements of the
Galois field $GF(2^q)$. We know that the elementary symmetric  polynomial $e_j$,  $1\leq j \leq k$, 
over the $n$-tuple $(\alpha_1,\alpha_2,\cdots, \alpha_n)$ are defined  as follows 
\begin{eqnarray}
e_1(\alpha_1,\alpha_2,\cdots, \alpha_n)&=&\sum_{1\leq j \leq n} \, \alpha_j\quad ,\\
\\
e_2(\alpha_1,\alpha_2,\cdots, \alpha_n)&=&\sum_{1\leq j<k \leq n} \, \alpha_j\, \alpha_k \quad ,\\
\\
e_k(\alpha_1,\alpha_2,\cdots, \alpha_n)&=&\sum_{1\leq j_1<j_2<\cdots <j_k \leq n} \, \alpha_{j_1}\cdots \alpha_{j_k} \quad .
\end{eqnarray}
In my research, I need an example of eight-tuple $(\alpha_1,\alpha_2,\cdots, \alpha_8)$ where $\alpha_i$, $1\leq i \leq 8$, be 
elements of the Galois field $GF(2^8)$ such that
$$
e_i(\alpha_1,\alpha_2,\cdots, \alpha_8)=0 \quad , \quad 1\leq i \leq 4 \quad . 
$$
I want to ask you to help me to obtain this example. If you find this example, please clarify with which primitive 
polynomial you have constructed the Galious field $GF(2^8)$. 
I wrote a Maple code for obtaining this example, but because of there are $256 \choose 8$ cases for choosing $\alpha_i$'s and
  limitation of my computer CPU and RAM, I could not find an example. The
JPG format of my Maple code is in the following form.
I would greatly appreciate for any assistance.
Edit:
Based on the efficient and excellent answers of Professor Robert Israel and Jyrki Lahtonen, I want to add an addition condition as follows
$$
e_i(\alpha_1,\alpha_2,\cdots, \alpha_8)=0 \quad , \quad 1\leq i \leq 4 \quad, \quad e_j(\alpha_1,\alpha_2,\cdots, \alpha_8)\neq0 \quad , \quad 5\leq j \leq 7\, .$$

With $GF(2^8)$ generated by a root $\beta$ of the polynomial $x^8+x^6+x^5+x^2+1$ over $GF(2)$, take
$$\eqalign{0,\; {\beta}^{3}+\beta,\;{\beta}^{7}+{\beta}^{5}+{\beta}^{2},\;{\beta}^{6}+{
\beta}^{5}+\beta+1,\;{\beta}^{6}+{\beta}^{5}+{\beta}^{3}+1,\;\cr{\beta}^{7}+{
\beta}^{5}+{\beta}^{3}+{\beta}^{2}+\beta,\;{\beta}^{7}+{\beta}^{6}+{
\beta}^{2}+\beta+1,\;{\beta}^{7}+{\beta}^{6}+{\beta}^{3}+{\beta}^{2}+1
}$$ 
Maple code:
P:= x^8+x^6+x^5+x^2+1:
alias(beta = RootOf(P)):
randc:= proc() add(rand(0..1)()*beta^j,j=0..7) end proc:
for iter from 1 do
  Q:= x^8 + add(randc()*x^j,j=1..3);
  F:= (Factors(Q,beta) mod 2)[2];
  if nops(F) = 8 then
     print(map(t -> subs(x=0,t[1]), F));
     break
  fi
od:

