On the Pfister Number of Quadratic Forms

TL;DR

This paper investigates the minimal number of 2-fold Pfister forms needed to express quadratic forms, providing explicit results for forms of dimension 6 with trivial discriminant over certain fields.

Contribution

It establishes the exact Pfister number for quadratic forms of dimension 6 with trivial discriminant and analyzes the general case over fields with specific properties.

Findings

Quadratic forms of dimension 6 with trivial discriminant require exactly 3 Pfister forms in certain cases.

Generic quadratic forms of even dimension n can be expressed as a sum of n-2 2-fold Pfister forms.

The minimal number of Pfister forms depends on the field's properties and the form's discriminant.

Abstract

The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n-2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.