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

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.