The spinor genus of the integral trace

TL;DR

This paper proves that for number fields of degree at least 3, the genus of their integral trace form contains only one spinor genus, with specific proportions for quadratic fields.

Contribution

It establishes the uniqueness of the spinor genus within the genus of the integral trace form for certain number fields and quantifies this property across quadratic fields.

Findings

The genus of the integral trace form contains only one spinor genus for degree ≥ 3 fields.

Approximately 43% of quadratic fields share this property.

The proportions are 29% for real quadratic and 58% for imaginary quadratic fields.

Abstract

Let $K$ be a number field of degree at least $3$. In this article we show that the genus of the integral trace form of $K$ contains only one spinor genus. Additionally we show that exactly $43%$ (resp. $29%$, resp. $58%$) of quadratic (resp. real quadratic, resp. imaginary quadratic) fields have the same property.