On number fields with equivalent integral trace forms

TL;DR

This paper investigates when different number fields can have equivalent integral trace forms, revealing conditions under which non-conjugated fields share these forms and proposing a conjecture for totally real quartic fields.

Contribution

The paper establishes that non-totally real number fields with the same signature and prime discriminant have equivalent trace forms, and conjectures a similar criterion for totally real quartic fields.

Findings

Non-totally real fields with same signature and prime discriminant have equivalent trace forms.

Conjecture: Totally real quartic fields with fundamental discriminant have equivalent trace zero forms iff they are conjugated.

Provides evidence supporting the conjecture for specific classes of number fields.

Abstract

Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.