Malle's Conjecture for Galois octic fields over $\mathbb Q$

TL;DR

This paper confirms Malle's conjecture for Galois octic fields with Galois group D_4 over Q, providing explicit asymptotics and demonstrating the proportionality constant's local mass structure, a first for non-abelian, non-symmetric families.

Contribution

It verifies the strong form of Malle's conjecture for D_4 octic fields and computes the asymptotic count, including the proportionality constant, for the first time in such non-abelian, non-symmetric cases.

Findings

Confirmed Malle's conjecture for D_4 octic fields.

Computed the asymptotic number of such fields by discriminant.

Showed the proportionality constant satisfies the Malle--Bhargava principle.

Abstract

We compute the asymptotic number of octic number fields whose Galois groups over $\mathbb Q$ are isomorphic to $D_4$, the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle's conjecture for such octic $D_4$-fields and obtain the constant of proportionality. Our result answers the question of whether a positive proportion of Galois octic extensions of $\mathbb Q$ have non-abelian Galois group in the negative. We further demonstrate that the constant of proportionality satisfies the Malle--Bhargava principle of being a product of local masses, despite the fact that this principle does {\em not} hold for discriminants of quartic $D_4$-fields. This is the first instance of asymptotics being recovered for a non-concentrated family of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, degree-$n$ $S_n$-fields for $n=3,4,5$, and degree-$6$ $S_3$-fields.