The Shape of cyclic number fields

TL;DR

This paper constructs a matrix representing the trace zero form of tame cyclic number fields, showing that their shape is determined solely by degree and discriminant, especially for real fields.

Contribution

It introduces a universal matrix for tame cyclic fields that encodes the shape from degree and discriminant alone, extending to real fields.

Findings

The matrix $A(rak{d})$ encodes the shape of tame cyclic number fields.

The shape is determined by degree and discriminant for these fields.

For real fields, the shape is also determined by degree and discriminant.

Abstract

Let $m>1$ and $\mathfrak{d} \neq 0$ be integers such that $v_{p}(\mathfrak{d}) \neq m$ for any prime $p$. We construct a matrix $A(\mathfrak{d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak{d}$ with the following property: For any tame $\mathbb{Z}/m\mathbb{Z}$-number field $K$ of discriminant $\mathfrak{d}$ the matrix $A(\mathfrak{d})$ represents the Gram matrix of the integral trace zero form of $K$. In particular, we have that the integral trace zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant