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Normal Quaternionic Matrices and Finitely Generated Witt Rings
Nico Lorenz, Alexander Schönert
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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 2ⁿ square classes in terms of a unique n×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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