$S_4$-quartics with Prescribed Norms

TL;DR

This paper investigates the distribution of $S_4$-quartic extensions of a number field where certain elements are norms, providing explicit local mass formulas and an algorithm for the remaining cases.

Contribution

It introduces a method to compute the density of $S_4$-quartic extensions with prescribed norms, including explicit formulas and an algorithm for complex cases.

Findings

Density expressed as product of local masses at all places

Explicit local mass formulas provided in most cases

Algorithm developed for computing remaining local masses

Abstract

Given a number field $k$ and a finitely generated subgroup $\mathcal{A} \subseteq k^*$, we study the distribution of $S_4$-quartic extensions of $k$ such that the elements of $\mathcal{A}$ are norms. We show that the density of such extensions is the product of so-called "local masses" at every place of $k$. We give these local masses explicitly in almost all cases and give an algorithm for computing the remaining cases.