= S4 criterion from the cubic resolvent of an irreducible quartic
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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.
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