Pfister Numbers over Rigid Fields

TL;DR

This paper investigates the minimal number of n-fold Pfister forms needed to represent certain quadratic forms over rigid fields, providing bounds and exact values where possible, thus advancing understanding of quadratic form decompositions.

Contribution

It introduces bounds and exact values for Pfister numbers over rigid fields, focusing on forms in the n-th power of the fundamental ideal, a novel contribution in quadratic form theory.

Findings

Computed upper bounds for Pfister numbers.

Determined exact Pfister numbers in specific cases.

Focused on rigid fields where binary anisotropic forms are limited.

Abstract

For certain types of quadratic forms lying in the n-th power of the fundamental ideal, we compute upper bounds and where possible exact values for the minimal number of general n-fold Pfister forms, that are needed to write the Witt class of that given form as the sum of the Witt classes of those n-fold Pfister forms. We restrict ourselves mostly to the case of so called rigid fields, i.e. fields in which binary anisotropic forms represent at most 2 square classes.