Normal Quaternionic Matrices and Finitely Generated Witt Rings

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. 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 satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

arXiv commentThe Python script along with documentation and a demo notebook can be found in the Zenodo repository 'Normal Quaternionic Matrices' https://doi.org/10.5281/zenodo.22873583