# An introduction to $\Gamma$-number fields

## TL;DR

This paper introduces a family of number fields where the spinor class of the integral trace uniquely determines the field's signature and discriminant, extending known results to odd degree Galois tame fields.

## Contribution

It defines a new family of number fields for which the spinor class of the integral trace fully characterizes the field, including odd degree Galois tame fields.

## Key findings

- The spinor class of the integral trace determines the signature and discriminant for the new family.
- Includes all odd degree Galois tame number fields within this family.
- The converse of quadratic form generalities holds for these fields.

## Abstract

It follows from generalities of quadratic forms that the spinor class of the integral trace of a number field determines the signature and the discriminant of the field. In this paper we define a family of number fields, that contains among others all odd degree Galois tame number fields, for which the converse is true. In other words, for a number field $K$ in such family we prove that the spinor class of the integral trace carries no more information about $K$ than the determinant and the signature do.