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arXiv · 2209.13579

Power-saving error terms for the number of $D_4$-quartic extensions over a number field ordered by discriminant

Alina Bucur·Alexandra Florea·Allechar Serrano López·Ila Varma
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Abstract

We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Kl\"uners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $D_4$-quartic extensions over a general base number field ordered by the norms of their relative discriminants. We also give an extensive overview of the history and development of number field asymptotics.

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Alina Bucur, Alexandra Florea, Allechar Serrano López, Ila Varma. 2022-09-27. Power-saving error terms for the number of $D_4$-quartic extensions over a number field ordered by discriminant. https://arxiv.org/abs/2209.13579

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