Number fields without universal quadratic forms of small rank exist in most degrees

TL;DR

This paper proves that for most degrees divisible by 2 or 3, there are infinitely many totally real number fields where universal quadratic forms must have arbitrarily large rank, highlighting limitations in quadratic form universality.

Contribution

It establishes the existence of infinitely many such fields in each degree divisible by 2 or 3, showing that small-rank universal quadratic forms are not possible in these cases.

Findings

Existence of infinitely many totally real fields with large-rank universal quadratic forms in degrees divisible by 2 or 3

Universal quadratic forms require arbitrarily large rank in these fields

Most degrees divisible by 2 or 3 contain such fields

Abstract

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.