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

DOI10.5802/JTNB.1206arXiv2009.09274MaRDI QIDQ6349451

Publication date: 19 September 2020

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

Mathematics Subject Classification ID

Arithmetic theory of algebraic function fields (11R58) Quadratic extensions (11R11) Cubic and quartic extensions (11R16) Density theorems (11R45)

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