Enumerating D 4 quartics and a Galois group bias over function fields
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 371-391.

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.

Nous donnons une formule asymptotique pour le nombre d’extensions quartiques du type D 4 et de discriminant donné d’un corps de fonctions en démontrant un résultat analogue à celui de Cohen, Diaz y Diaz et Olivier pour les corps de nombres mais avec un meilleur terme d’erreur. Nous étudions aussi la densité relative des extensions quartiques des types D 4 et S 4 d’un corps de fonctions. Nous montrons que sous des hypothèses faibles, le nombre d’extensions quartiques du type D 4 peut largement dépasser le nombre d’extensions quartiques du type S 4 .

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1206
Classification: 11R45, 11R11, 11R16, 11R58
Keywords: Field counting, function fields, Galois theory, polynomials
Daniel Keliher 1

1 Tufts University 177 College Ave. Medford, MA, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Daniel Keliher. Enumerating $D_4$ quartics and a Galois group bias over function fields. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 371-391. doi : 10.5802/jtnb.1206. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1206/

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