Algèbre
La forme trace d'une algèbre simple centrale de degré 4
[The trace form of a central simple algebra of degree 4]
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 83-87.

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 q2 and a 4-fold Pfister form q4 which are uniquely determined by A. The form q2 is the norm form of the quaternion algebra Brauer-equivalent to AkA, and q4 is hyperbolic if and only if A is a symbol algebra.

Soit k un corps de caractéristique ≠2 dans lequel −1 est un carré, et soit A une algèbre simple centrale sur k de degré 4. La forme trace de A s'écrit de façon unique comme une somme (au sens de Witt) q2+q4, où q2 (resp. q4) est une 2-forme de Pfister (resp. une 4-forme de Pfister). On a q4=0 si et seulement si A est cyclique, et q2=0 si et seulement si 2.[A]=0 dans Br(k).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.11.002
Rost, Markus 1; Serre, Jean-Pierre 2; Tignol, Jean-Pierre 3

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Allemagne
2 Collège de France, 3, rue d'Ulm, 75005 Paris, France
3 Département de mathématique, université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgique
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Rost, Markus; Serre, Jean-Pierre; Tignol, Jean-Pierre. La forme trace d'une algèbre simple centrale de degré 4. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 83-87. doi : 10.1016/j.crma.2005.11.002. http://www.numdam.org/articles/10.1016/j.crma.2005.11.002/

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