Enumerative Galois theory for cubics and quartics

Abstract

We show that there are $O_{\epsilon }(H^{1.5+\epsilon })$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2(\log H{)}^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2\log H)$, $O(H^2\log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.