Counting Cubic Extensions with given Quadratic Resolvent

TL;DR

This paper provides an explicit asymptotic count of cubic extensions over a number field with a specified quadratic subfield, using Kummer theory, and analyzes the error term in detail.

Contribution

It offers a new explicit asymptotic formula for counting cubic extensions with a given quadratic resolvent, including a detailed error analysis.

Findings

Asymptotic formula for cubic extensions with quadratic resolvent

Explicit error term bound of O(X^α) with α<1

Application of Kummer theory to counting problems

Abstract

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha})$, for an explicit $\alpha<1$.