# The trace form over cyclic number fields

## TL;DR

This paper generalizes a classical result relating the isometry class of the integral trace to the discriminant from prime degree cyclic fields to arbitrary tame cyclic fields, providing explicit Gram matrix descriptions.

## Contribution

It extends the known relationship between trace form isometry classes and discriminants to all tame cyclic number fields of any degree, with explicit Gram matrix formulas.

## Key findings

- Isometry class determined by discriminant for tame cyclic fields
- Explicit Gram matrix formulas in terms of discriminant
- Generalization from prime degree to arbitrary tame cyclic fields

## Abstract

In the mid 80's Conner and Perlis showed that for cyclic number fields of prime degree $p$ the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.