Characteristic-two cubic resolvent
= Characteristic-two cubic resolvent
Let $x_1,x_2,x_3$ be the roots of the separable polynomial $T^3+aT+b$ in characteristic two. The cyclic sums
$$
r=x_1^2x_2+x_2^2x_3+x_3^2x_1,
\qquad
s=x_2^2x_1+x_3^2x_2+x_1^2x_3
$$
are the distinct roots of
$$
T^2+bT+a^3+b^2.
$$
Even root permutations fix $r,s$, while odd permutations interchange them. The cubic Galois group is therefore contained in $A_3$ exactly when this quadratic splits over the base field.