An improved error term for counting D₄-quartic fields

2023/07/27 by Kevin J. McGown, McGown, Kevin J., Amanda Tucker +1 · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2307.14564

Abstract

We prove that the number of quartic fields K with discriminant |Δ_K|≤ X whose Galois closure is D₄ equals CX+O(X^5/8+ε), improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

Cited by

• Counting C₂ \wr S₄ fields with a power saving error term
• A power-saving error term in counting C₂ \wr H extensions of an arbitrary base field parametrized by discriminants
• Non-vanishing of Artin L-functions associated with D₄-quartic function fields ordered by conductor

Related

• Enumerating D₄ Quartics and a Galois Group Bias Over Function Fields
• Lower bounds for counting A₄-quartic fields
• Power-saving error terms for the number of D₄-quartic extensions over a number field ordered by discriminant
• The number of quartic D₄-fields ordered by conductor
• Progress Towards Counting D₅ Quintic Fields