Enumerating $D_4$ Quartics and a Galois Group Bias Over Function Fields

TL;DR

This paper derives an asymptotic count for $D_4$ quartic extensions over function fields, improves error estimates, and reveals a bias favoring $D_4$ over $S_4$ extensions under certain conditions.

Contribution

It provides a precise asymptotic formula for $D_4$ quartic extensions over function fields with enhanced error bounds and analyzes the relative density compared to $S_4$ extensions.

Findings

Asymptotic formula for $D_4$ quartic extensions with strong error term

Demonstrates $D_4$ extensions can significantly outnumber $S_4$ extensions

Shows a bias towards $D_4$ extensions under mild conditions

Abstract

We give an asymptotic formula for the number of $D_4$ quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of $D_4$ and $S_4$ quartic extensions of a function field and show that with mild conditions, the number of $D_4$ quartic extensions can far exceed the number of $S_4$ quartic extensions