Fields where torsion forms decompose

TL;DR

This paper proves that over certain real fields, torsion quadratic forms decompose into sums of 2-dimensional torsion forms, based on a broader study of weakly isotropic forms over specific fields.

Contribution

It introduces a new decomposition result for torsion quadratic forms over particular real fields, extending understanding of form behavior over valued and function fields.

Findings

Quadratic torsion forms decompose into 2-dimensional torsion forms over specified fields.

Decomposition results are derived from a general study of weakly isotropic forms.

The study applies to henselian valued fields and function fields in one variable.

Abstract

Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more general study of weakly isotropic forms over henselian valued fields and over function fields in one variable.