Enumerative Galois theory for cubics and quartics.

We show that there are order of magnitude $H^2 (\log H)^2$ monic quartic polynomials with integer coefficients having box height at most $H$ whose Galois group is $D_4$. Further, we prove that the corresponding number of $V_4$ and $C_4$ quartics is $O(H^2 \log H)$. We also show that the count for $A_4$ quartics is $O(H^{2.95})$. Finally, we prove that the corresponding count for $A_3$ cubics is $O_\varepsilon(H^{1.5+\varepsilon})$. Our work establishes that irreducible non-$S_4$ quartics are less numerous than reducible quartics, and similarly in the cubic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.