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Abstract
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A theorem of Pfister asserts that every
-dimensional
quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic
different from
decomposes as a tensor product of a binary quadratic form and a
-dimensional quadratic form
with trivial discriminant. Our main result extends Pfister’s result to orthogonal involutions: every central
simple algebra of degree
with orthogonal involution of trivial discriminant and trivial Clifford invariant
decomposes into a tensor product of a quaternion algebra and a central simple algebra
of degree
with orthogonal involutions. This decomposition is used to establish a criterion
for the existence of orthogonal involutions with trivial invariants on algebras of degree
, and to calculate the
-invariant of the involution
if the algebra has index
.
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Keywords
algebra with involution, hermitian form, cohomological
invariant
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Mathematical Subject Classification 2010
Primary: 11E72
Secondary: 11E81, 16W10
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Milestones
Received: 8 May 2019
Revised: 11 July 2019
Accepted: 25 July 2019
Published: 18 January 2020
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