 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.