On the Arithmetic determination of the trace

TL;DR

This paper provides a numerical criterion based on ramification behavior to determine when the integral trace of a number field is isometric to that of another, with specific results for cubic fields of negative discriminant.

Contribution

It introduces a ramification-based numerical criterion for trace isometry between number fields, extending previous results for cubic fields of positive discriminant.

Findings

A criterion depending only on ramification behavior for trace isometry.

For cubic fields of negative discriminant, trace isometry is equivalent to discriminant equality.

Abstract

Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is isometric to the integral trace of another number field $L$. As a byproduct of our proofs here, and in contrast with our previous results for cubic fields of positive discriminant, we show that for cubic fields of negative discriminant isometry between integral traces is equivalent to equality of discriminants.