# The integral trace form as a complete invariant for real $S_n$ number fields

## TL;DR

This paper demonstrates that for certain real $S_n$ number fields with specific ramification bounds, the integral trace uniquely characterizes the field, extending previous understanding beyond cyclic and non-totally real cases.

## Contribution

It establishes that the integral trace is a complete invariant for specific real $S_n$ number fields and provides an explicit description of their isometry groups.

## Key findings

- Integral trace characterizes certain real $S_n$ fields uniquely.
- Explicit description of isometry groups for these fields.
- Applicable to fields with square-free different ideals.

## Abstract

In the past the first named author has studied to what extent the integral trace can characterize a number field beyond what the discriminant does. The cases of cyclic number fields and non-totally real fields are more or less settled, concluding that for such fields the integral trace does not always characterize the field. In this paper we show that the integral trace is a complete invariant for degree $n$, $S_n$ real number number fields that satisfy certain ramification bound. Among the real $S_n$ fields that our results cover, there are those of square free different ideal. Moreover, for such fields we find an explicit description of the isometry group of the integral trace.