Cubic resolvent of a quartic

For a depressed quartic $x^4+p{x}^2+qx+r$ with roots ${\alpha }_1,\dots ,{\alpha }_4$, the three pair-partition quantities ${\alpha }_1{\alpha }_2+{\alpha }_3{\alpha }_4$ and its two analogues are the roots ofThe Galois action on these roots is the quotient of its action on the four quartic roots by the Klein four-group.

If an irreducible quartic over a field of characteristic zero has a cubic resolvent of a quartic whose Galois group of a polynomial is $S_3$, then the quartic has Galois group $S_4$. Irreducibility places the quartic group among the transitive subgroups of the symmetric group on four points, while the action on its three root pairings is surjective onto $S_3$. Of the possibilities $C_4,V_4,D_8,A_4,S_4$, only $S_4$ has this image.