# Triple Linkage of Quadratic Pfister Forms

## TL;DR

This paper establishes conditions under which certain quadratic forms and quaternion algebras over fields have common factors or subfields, linking algebraic properties to the u-invariant of the field.

## Contribution

It proves that the common factor condition for three quadratic Pfister forms implies the vanishing of a specific power of the fundamental ideal, and characterizes the u-invariant for fields with shared maximal subfields.

## Key findings

- If three quadratic n-fold Pfister forms share a common factor, then $I_q^{n+1} F=0.
- If three quaternion algebras over a field share a maximal subfield, then the u-invariant is 0, 2, or 4.
- For nonreal fields with u-invariant 4, three quaternion algebras share a maximal subfield.

## Abstract

Given a field $F$ of characteristic 2, we prove that if every three quadratic $n$-fold Pfister forms have a common quadratic $(n-1)$-fold Pfister factor then $I_q^{n+1} F=0$. As a result, we obtain that if every three quaternion algebras over $F$ share a common maximal subfield then $u(F)$ is either $0,2$ or $4$. We also prove that if $F$ is a nonreal field with $\operatorname{char}(F) \neq 2$ and $u(F)=4$, then every three quaternion algebras share a common maximal subfield.