Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields

TL;DR

This paper investigates the limitations on universal quadratic lattices and indecomposable elements in real multiquadratic fields, providing density zero results and detailed structural insights for biquadratic fields.

Contribution

It establishes upper bounds on the number of real multiquadratic fields with certain universal lattice properties and analyzes indecomposable elements in biquadratic fields, including explicit computations.

Findings

Density zero results for fields with universal lattices of fixed rank

Structural characterization of indecomposable elements in real biquadratic fields

Explicit systems of indecomposable elements computed for specific families

Abstract

We establish an upper bound on the number of real multiquadratic fields that admit a universal quadratic lattice of a given rank, or contain a given amount of indecomposable elements modulo totally positive units, obtaining density zero statements. We also study the structure of indecomposable elements in real biquadratic fields, and compute a system of indecomposable elements modulo totally positive units for some families of real biquadratic fields.