Supreme Torsion Forms

TL;DR

This paper investigates special types of formally real fields with a universal anisotropic torsion form, exploring their invariants, Witt rings, and constructing examples, drawing parallels to supreme Pfister forms.

Contribution

It introduces the concept of supreme torsion forms in formally real, non-pythagorean fields and develops a related theoretical framework with new examples.

Findings

Existence of fields with a universal anisotropic torsion form

Consequences for invariants and Witt rings of such fields

Examples where Pythagoras number behaves like the level

Abstract

We study formally real, non-pythagorean fields which have an anisotropic torsion form that contains every anisotropic torsion form as a subform. We obtain consequences for certain invariants and the Witt ring of such fields and construct examples. We obtain a theory analogous to the theory of supreme Pfister forms introduced by Karim Becher and see examples in which the Pythagoras number for formally real fields behaves like the level for nonreal fields.