 Solution

Denote the three stated pair sums by $r_1,r_2,r_3$. They are pairwise distinct: for example,and the other differences factor similarly. Every Galois automorphism fixes each $r_i\in F$, and hence fixes each of the three corresponding partitions of four roots into two pairs. The kernel of the permutation action $S_4\to S_3$ on these partitions isThus $G$ embeds in the Klein four-group. By part (a), $4$ divides $|G|$, and the containing group has order four. ConsequentlyThis Split resolvent criterion for a Klein four Galois group uses the cubic resolvent of a quartic. The pair-sum differences remain nonzero in characteristic two, so this proof does not need the missing characteristic assumption from part (c).