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arXiv · 2505.14485

Normal Quaternionic Matrices and Finitely Generated Witt Rings

Nico Lorenz·Alexander Schönert
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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.

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Nico Lorenz, Alexander Schönert. 2025-05-20. Normal Quaternionic Matrices and Finitely Generated Witt Rings. https://arxiv.org/abs/2505.14485

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