Extending bounds on minimal ranks of universal quadratic lattices to larger number fields

TL;DR

This paper improves methods for establishing lower bounds on the minimal rank of universal quadratic lattices over larger number fields, using Galois theory to relate properties across degrees.

Contribution

It extends Kala's technique by analyzing subfield structures via Galois theory, showing the existence of fields with certain minimal ranks in higher degrees.

Findings

If such fields exist in degree d, they also exist in degree kd for all k≥3.

The approach translates field extension problems into group-theoretic problems.

Provides a framework for constructing larger degree fields with prescribed minimal ranks.

Abstract

There exist numerous results in the literature proving that within certain families of totally real number fields, the minimal rank of a universal quadratic lattice over such a field can be arbitrarily large. Kala introduced a technique of extending such results to larger fields -- e.g. from quadratic fields to fields of arbitrary even degree -- under some conditions. We present improvements to this technique by investigating the structure of subfields within composita of number fields, using basic Galois theory to translate this into a group-theoretic problem. In particular, we show that if totally real number fields with minimal rank of a universal lattice $\geq r$ exist in degree $d$, then they also exist in degree $kd$ for all $k\geq3$.