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

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 $k\geq3$.