Enumerative Galois theory for cubics and quartics

Abstract

We show that there are Oε(H1.5+ε) monic, cubic polynomials with integer coefficients bounded by H in absolute value whose Galois group is A3. We also show that the order of magnitude for D4 quartics is H2(log⁡H)2, and that the respective counts for A4, V4, C4 are O(H2.91), O(H2log⁡H), O(H2log⁡H). Our work establishes that irreducible non-S3 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.