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The trace form of a central simple algebra of degree 4. (La forme trace d’une algèbre simple centrale de degré 4.) (French) Zbl 1110.16014

Summary: Let \(k\) be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple \(k\)-algebra \(A\) of degree 4 decomposes in the Witt group of \(k\) as the sum of a 2-fold Pfister form \(q_2\) and a 4-fold Pfister form \(q_4\) which are uniquely determined by \(A\). The form \(q_2\) is the norm form of the quaternion algebra Brauer-equivalent to \(A\otimes_kA\), and \(q_4\) is hyperbolic if and only if \(A\) is a symbol algebra.

MSC:

16K20 Finite-dimensional division rings
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields
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References:

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