Normal Quaternionic Matrices and Finitely Generated Witt Rings

TL;DR

This paper introduces a novel method to analyze abstract Witt rings using quaternionic matrices, verifying the Elementary Type Conjecture for rings with up to 128 square classes through computational enumeration.

Contribution

It develops a matrix-based description of abstract Witt rings and confirms the Elementary Type Conjecture for small cases via computational search.

Findings

Verified the Elementary Type Conjecture for up to 7 square classes.

Developed a unique matrix representation for Witt rings with 2^n classes.

Confirmed the conjecture's validity through exhaustive computational enumeration.

Abstract

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group, also verifying a certain case of the Arason-Pfister Hauptsatz in this setting. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix. Via computational search, we find all these matrices for $n$ up to $7$. All obtained results affirm the Elementary Type Conjecture.