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Zbl 1110.16015
Tignol, Jean-Pierre
The second trace form of a central simple algebra of degree 4 of characteristic 2. (La forme seconde trace d'une algèbre simple centrale de degré 4 de caractéristique 2.) (French)
[J] C. R., Math., Acad. Sci. Paris 342, No. 2, 89-92 (2006). ISSN 1631-073X

Summary: Let $A$ be a central simple algebra of degree 4 over a field $k$ of characteristic 2 and let $q_A$ be the quadratic form on $A$ given by the second coefficient of the reduced characteristic polynomial. We show that $A$ uniquely determines a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ such that $q_A=[1,1]+q_2+q_4$ in the Witt group of $k$, where $[1,1]$ is the form $x^2+xy+y^2$. 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 cyclic.

MSC 2000:: *16K20 Finite-dimensional division rings
11E81 Algebraic theory of quadratic forms
11E04 Quadratic forms over general fields

Keywords: quadratic forms; central simple algebras; Witt groups; Pfister forms; norm forms; quaternion algebras; characteristic polynomials; cyclic algebras

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