I am trying to show for some complex numbers `$l_j , 1 ≤ j ≤ 5$
$∆(x_1, x_2, x_3)=l_1σ^2_{3,3} + l_2σ_{3,1}σ_{3,2}σ_{3,3} + l_3σ^3_{3,1}σ_{3,3}+ l_4σ^3_{3,2} + l_5σ^2_{3,1}σ^2_{3,2}$.
Second, by using the polynomial $x^ 3 − 1$, want to show that $l_1 = −27$.
I know the discriminant $∆_{y3+py+q} = −4p^3 − 27q^ 2$ and P is a symmetric polynomial if for any permutation $σ$ of the subscripts $1, 2, \ldots, n$ one has $F(X_{σ(1)},\ldots, X_{σ(n)}) = P(X_1,\ldots, X_n)$. But I am not sure how to proceed in the above example.