Integral trace forms associated to cubic extensions

TL;DR

This paper investigates whether the trace form can uniquely identify cubic fields, showing it does so for totally real fields of fundamental discriminant using Bhargava's class group and composition laws.

Contribution

It introduces a new refinement of the discriminant via the trace form and demonstrates its completeness for totally real cubic fields with fundamental discriminant.

Findings

Trace form determines cubic fields in certain cases

Existence of an element in Bhargava's class group linked to the trace form

Complete invariant property for totally real cubic fields

Abstract

Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form $q_K:\text{tr}_{K/\mathbb{Q}}(x^2)|_{O^{0}_{K}}$ as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element $T_K$ in Bhargava's class group $\Cl(\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}; -3d)$ such that $q_K$ is completely determined by $T_K$. By using one of Bhargava's composition laws, we show that $q_K$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant