# A proof of a conjecture on trace-zero forms and shapes of number fields

## TL;DR

This paper proves a conjecture that totally real quartic fields are uniquely determined by their trace-zero form, extending the result to higher degrees under certain conditions, using Bhargava's parametrization and previous work on trace forms.

## Contribution

It proves a conjecture for quartic fields using Bhargava's parametrization and extends the result to higher degrees with specific cyclicity conditions on units.

## Key findings

- Totally real quartic fields are determined by their trace-zero form.
- The shape is a complete invariant for certain higher degree fields.
- Extension of the result to degrees where $(Z/nZ)^*$ is cyclic.

## Abstract

In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog statement for cubic fields which was proved using Bhargava's higher composition laws on cubes. Here, using Bhargava's parametrization of quartic fields we prove the conjecture by generalizing the ideas used in the cubic case. Since at the moment, for arbitrary degrees, there is nothing like Bhargava's parametrizations we cannot deal with degrees $n > 5$ in a similar fashion. Nevertheless, using some of our previous work on trace forms we generalize this result to higher degrees; we show that if $n \ge 3$ is an integer such that $(\mathbb{Z}/n\mathbb{Z})^{*}$ is a cyclic group, then the shape is a complete invariant for totally real degree $n$ number fields with fundamental discriminant.