Decomposable and indecomposable algebras of degree 8 and exponent 2

Barry, Demba
(2014) Mathematische Zeitschrift — Vol. 276, n° 3, p. 1113-1132 (2014)

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  • Barry, DembaUCLouvain
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Abstract
We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F(√a) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F. This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field M(t) where the u-invariant of M is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.
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Barry, D. (2014). Decomposable and indecomposable algebras of degree 8 and exponent 2. Mathematische Zeitschrift, 276(3), 1113-1132. https://doi.org/10.1007/s00209-013-1236-8 (Original work published 2014)