English

An improved error term for counting $D_4$-quartic fields

Number Theory 2024-05-07 v4

Abstract

We prove that the number of quartic fields KK with discriminant ΔKX|\Delta_K|\leq X whose Galois closure is D4D_4 equals CX+O(X5/8+ε)CX+O(X^{5/8+\varepsilon}), 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.

Cite

@article{arxiv.2307.14564,
  title  = {An improved error term for counting $D_4$-quartic fields},
  author = {Kevin J. McGown and Amanda Tucker},
  journal= {arXiv preprint arXiv:2307.14564},
  year   = {2024}
}

Comments

17 pages, 1 figure

R2 v1 2026-06-28T11:41:23.565Z