Enumerative Galois theory for cubics and quartics

TL;DR

This paper provides asymptotic counts for monic cubic and quartic polynomials with integer coefficients, classifying their Galois groups, and confirms a 1936 conjecture about the rarity of certain irreducible polynomials.

Contribution

It establishes the first asymptotic bounds for the number of cubics and quartics with specific Galois groups, confirming van der Waerden's conjecture.

Findings

Number of cubic polynomials with Galois group A_3 is O(H^{1.5+ε})

Quartic polynomials with Galois group D_4 are counted as H^2 (log H)^2

Irreducible non-S_3 cubics are less common than reducible ones

Abstract

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ 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.